INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV:
LOG-VOLUME
COMPUTATIONS
AND
SET-THEORETIC
FOUNDATIONS
Shinichi
Mochizuki
April
2020
Abstract.
The
present
paper
forms
the
fourth
and
final
paper
in
a
series
of
papers
concerning
“inter-universal
Teichmüller
theory”.
In
the
first
three
papers
of
the
series,
we
introduced
and
studied
the
theory
surrounding
the
log-
theta-lattice,
a
highly
non-commutative
two-dimensional
diagram
of
“miniature
models
of
conventional
scheme
theory”,
called
Θ
±ell
NF-Hodge
theaters,
that
were
associated,
in
the
first
paper
of
the
series,
to
certain
data,
called
initial
Θ-data.
This
data
includes
an
elliptic
curve
E
F
over
a
number
field
F
,
together
with
a
prime
number
l
≥
5.
Consideration
of
various
properties
of
the
log-theta-lattice
led
naturally
to
the
establishment,
in
the
third
paper
of
the
series,
of
multiradial
algorithms
for
constructing
“splitting
monoids
of
LGP-monoids”.
Here,
we
recall
that
“multiradial
algorithms”
are
algorithms
that
make
sense
from
the
point
of
view
of
an
“alien
arithmetic
holomorphic
structure”,
i.e.,
the
ring/scheme
structure
of
a
Θ
±ell
NF-Hodge
theater
related
to
a
given
Θ
±ell
NF-Hodge
theater
by
means
of
a
non-ring/scheme-theoretic
horizontal
arrow
of
the
log-theta-lattice.
In
the
present
paper,
estimates
arising
from
these
multiradial
algorithms
for
splitting
monoids
of
LGP-monoids
are
applied
to
verify
various
diophantine
results
which
imply,
for
instance,
the
so-called
Vojta
Conjecture
for
hyperbolic
curves,
the
ABC
Conjecture,
and
the
Szpiro
Conjecture
for
elliptic
curves.
Finally,
we
examine
—
albeit
from
an
extremely
naive/non-expert
point
of
view!
—
the
foundational/set-
theoretic
issues
surrounding
the
vertical
and
horizontal
arrows
of
the
log-theta-lattice
by
introducing
and
studying
the
basic
properties
of
the
notion
of
a
“species”,
which
may
be
thought
of
as
a
sort
of
formalization,
via
set-theoretic
formulas,
of
the
intuitive
notion
of
a
“type
of
mathematical
object”.
These
foundational
issues
are
closely
related
to
the
central
role
played
in
the
present
series
of
papers
by
various
results
from
absolute
anabelian
geometry,
as
well
as
to
the
idea
of
gluing
together
distinct
models
of
conventional
scheme
theory,
i.e.,
in
a
fashion
that
lies
outside
the
framework
of
conventional
scheme
theory.
Moreover,
it
is
precisely
these
foundational
issues
surrounding
the
vertical
and
horizontal
arrows
of
the
log-theta-lattice
that
led
naturally
to
the
introduction
of
the
term
“inter-universal”.
Contents:
Introduction
§0.
Notations
and
Conventions
§1.
Log-volume
Estimates
§2.
Diophantine
Inequalities
§3.
Inter-universal
Formalism:
the
Language
of
Species
Typeset
by
AMS-TEX
1
2
SHINICHI
MOCHIZUKI
Introduction
The
present
paper
forms
the
fourth
and
final
paper
in
a
series
of
papers
concern-
ing
“inter-universal
Teichmüller
theory”.
In
the
first
three
papers,
[IUTchI],
[IUTchII],
and
[IUTchIII],
of
the
series,
we
introduced
and
studied
the
theory
sur-
rounding
the
log-theta-lattice
[cf.
the
discussion
of
[IUTchIII],
Introduction],
a
highly
non-commutative
two-dimensional
diagram
of
“miniature
models
of
con-
ventional
scheme
theory”,
called
Θ
±ell
NF-Hodge
theaters,
that
were
associated,
in
the
first
paper
[IUTchI]
of
the
series,
to
certain
data,
called
initial
Θ-data.
This
data
includes
an
elliptic
curve
E
F
over
a
number
field
F
,
together
with
a
prime
number
l
≥
5
[cf.
[IUTchI],
§I1].
Consideration
of
various
properties
of
the
log-
theta-lattice
leads
naturally
to
the
establishment
of
multiradial
algorithms
for
constructing
“splitting
monoids
of
LGP-monoids”
[cf.
[IUTchIII],
Theorem
A].
Here,
we
recall
that
“multiradial
algorithms”
[cf.
the
discussion
of
the
Intro-
ductions
to
[IUTchII],
[IUTchIII]]
are
algorithms
that
make
sense
from
the
point
of
view
of
an
“alien
arithmetic
holomorphic
structure”,
i.e.,
the
ring/scheme
structure
of
a
Θ
±ell
NF-Hodge
theater
related
to
a
given
Θ
±ell
NF-Hodge
theater
by
means
of
a
non-ring/scheme-theoretic
horizontal
arrow
of
the
log-theta-lattice.
In
the
final
portion
of
[IUTchIII],
by
applying
these
multiradial
algorithms
for
split-
ting
monoids
of
LGP-monoids,
we
obtained
estimates
for
the
log-volume
of
these
LGP-monoids
[cf.
[IUTchIII],
Theorem
B].
In
the
present
paper,
these
estimates
will
be
applied
to
verify
various
diophantine
results.
In
§1
of
the
present
paper,
we
start
by
discussing
various
elementary
estimates
for
the
log-volume
of
various
tensor
products
of
the
modules
obtained
by
applying
the
p-adic
logarithm
to
the
local
units
—
i.e.,
in
the
terminology
of
[IUTchIII],
“tensor
packets
of
log-shells”
[cf.
the
discussion
of
[IUTchIII],
Introduction]
—
in
terms
of
various
well-known
invariants,
such
as
differents,
associated
to
a
mixed-
characteristic
nonarchimedean
local
field
[cf.
Propositions
1.1,
1.2,
1.3,
1.4].
We
then
discuss
similar
—
but
technically
much
simpler!
—
log-volume
estimates
in
the
case
of
complex
archimedean
local
fields
[cf.
Proposition
1.5].
After
review-
ing
a
certain
classical
estimate
concerning
the
distribution
of
prime
numbers
[cf.
Proposition
1.6],
as
well
as
some
elementary
general
nonsense
concerning
weighted
averages
[cf.
Proposition
1.7]
and
well-known
elementary
facts
concerning
elliptic
curves
[cf.
Proposition
1.8],
we
then
proceed
to
compute
explicitly,
in
more
elemen-
tary
language,
the
quantity
that
was
estimated
in
[IUTchIII],
Theorem
B.
These
computations
yield
a
quite
strong/explicit
diophantine
inequality
[cf.
Theorem
1.10]
concerning
elliptic
curves
that
are
in
“sufficiently
general
position”,
so
that
one
may
apply
the
general
theory
developed
in
the
first
three
papers
of
the
series.
In
§2
of
the
present
paper,
after
reviewing
another
classical
estimate
concern-
ing
the
distribution
of
prime
numbers
[cf.
Proposition
2.1,
(ii)],
we
then
proceed
to
apply
the
theory
of
[GenEll]
to
reduce
various
diophantine
results
concerning
an
arbitrary
elliptic
curve
over
a
number
field
to
results
of
the
type
obtained
in
Theorem
1.10
concerning
elliptic
curves
that
are
in
“sufficiently
general
posi-
tion”
[cf.
Corollary
2.2].
This
reduction
allows
us
to
derive
the
following
result
[cf.
Corollary
2.3],
which
constitutes
the
main
application
of
the
“inter-universal
Teichmüller
theory”
developed
in
the
present
series
of
papers.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
Theorem
A.
3
(Diophantine
Inequalities)
Let
X
be
a
smooth,
proper,
geomet-
def
rically
connected
curve
over
a
number
field;
D
⊆
X
a
reduced
divisor;
U
X
=
X\D;
d
a
positive
integer;
∈
R
>0
a
positive
real
number.
Write
ω
X
for
the
canon-
ical
sheaf
on
X.
Suppose
that
U
X
is
a
hyperbolic
curve,
i.e.,
that
the
degree
of
the
line
bundle
ω
X
(D)
is
positive.
Then,
relative
to
the
notation
of
[GenEll]
[reviewed
in
the
discussion
preceding
Corollary
2.2
of
the
present
paper],
one
has
an
inequality
of
“bounded
discrepancy
classes”
ht
ω
X
(D)
(1
+
)(log-diff
X
+
log-cond
D
)
of
functions
on
U
X
(Q)
≤d
—
i.e.,
the
function
(1
+
)(log-diff
X
+
log-cond
D
)
−
ht
ω
X
(D)
is
bounded
below
by
a
constant
on
U
X
(Q)
≤d
[cf.
[GenEll],
Definition
1.2,
(ii),
as
well
as
Remark
2.3.1,
(ii),
of
the
present
paper].
Thus,
Theorem
A
asserts
an
inequality
concerning
the
canonical
height
[i.e.,
“ht
ω
X
(D)
”],
the
logarithmic
different
[i.e.,
“log-diff
X
”],
and
the
logarithmic
conduc-
tor
[i.e.,
“log-cond
D
”]
of
points
of
the
curve
U
X
valued
in
number
fields
whose
extension
degree
over
Q
is
≤
d
.
In
particular,
the
so-called
Vojta
Conjecture
for
hyperbolic
curves,
the
ABC
Conjecture,
and
the
Szpiro
Conjecture
for
elliptic
curves
all
follow
as
special
cases
of
Theorem
A.
We
refer
to
[Vjt]
for
a
detailed
exposition
of
these
conjectures.
Finally,
in
§3,
we
examine
—
albeit
from
an
extremely
naive/non-expert
point
of
view!
—
certain
foundational
issues
underlying
the
theory
of
the
present
se-
ries
of
papers.
Typically
in
mathematical
discussions
[i.e.,
by
mathematicians
who
are
not
equipped
with
a
detailed
knowledge
of
the
theory
of
foundations!]
—
such
as,
for
instance,
the
theory
developed
in
the
present
series
of
papers!
—
one
de-
fines
various
“types
of
mathematical
objects”
[i.e.,
such
as
groups,
topological
spaces,
or
schemes],
together
with
a
notion
of
“morphisms”
between
two
partic-
ular
examples
of
a
specific
type
of
mathematical
object
[i.e.,
morphisms
between
groups,
between
topological
spaces,
or
between
schemes].
Such
objects
and
mor-
phisms
[typically]
determine
a
category.
On
the
other
hand,
if
one
restricts
one’s
attention
to
such
a
category,
then
one
must
keep
in
mind
the
fact
that
the
structure
of
the
category
—
i.e.,
which
consists
only
of
a
collection
of
objects
and
morphisms
satisfying
certain
properties!
—
does
not
include
any
mention
of
the
various
sets
and
conditions
satisfied
by
those
sets
that
give
rise
to
the
“type
of
mathematical
object”
under
consideration.
For
instance,
the
data
consisting
of
the
underlying
set
of
a
group,
the
group
multiplication
law
on
the
group,
and
the
properties
sat-
isfied
by
this
group
multiplication
law
cannot
be
recovered
[at
least
in
an
a
priori
sense!]
from
the
structure
of
the
“category
of
groups”.
Put
another
way,
although
the
notion
of
a
“type
of
mathematical
object”
may
give
rise
to
a
“category
of
such
objects”,
the
notion
of
a
“type
of
mathematical
object”
is
much
stronger
—
in
the
sense
that
it
involves
much
more
mathematical
structure
—
than
the
notion
of
a
cat-
egory.
Indeed,
a
given
“type
of
mathematical
object”
may
have
a
very
complicated
internal
structure,
but
may
give
rise
to
a
category
equivalent
to
a
one-morphism
category
[i.e.,
a
category
with
precisely
one
morphism];
in
particular,
in
such
cases,
the
structure
of
the
associated
category
does
not
retain
any
information
of
inter-
est
concerning
the
internal
structure
of
the
“type
of
mathematical
object”
under
consideration.
4
SHINICHI
MOCHIZUKI
In
Definition
3.1,
(iii),
we
formalize
this
intuitive
notion
of
a
“type
of
mathe-
matical
object”
by
defining
the
notion
of
a
species
as,
roughly
speaking,
a
collection
of
set-theoretic
formulas
that
gives
rise
to
a
category
in
any
given
model
of
set
the-
ory
[cf.
Definition
3.1,
(iv)],
but,
unlike
any
specific
category
[e.g.,
of
groups,
etc.]
is
not
confined
to
any
specific
model
of
set
theory.
In
a
similar
vein,
by
working
with
collections
of
set-theoretic
formulas,
one
may
define
a
species-theoretic
ana-
logue
of
the
notion
of
a
functor,
which
we
refer
to
as
a
mutation
[cf.
Definition
3.3,
(i)].
Given
a
diagram
of
mutations,
one
may
then
define
the
notion
of
a
“mutation
that
extracts,
from
the
diagram,
a
certain
portion
of
the
types
of
mathematical
objects
that
appear
in
the
diagram
that
is
invariant
with
respect
to
the
mutations
in
the
diagram”;
we
refer
to
such
a
mutation
as
a
core
[cf.
Definition
3.3,
(v)].
One
fundamental
example,
in
the
context
of
the
present
series
of
papers,
of
a
diagram
of
mutations
is
the
usual
set-up
of
[absolute]
anabelian
geometry
[cf.
Example
3.5
for
more
details].
That
is
to
say,
one
begins
with
the
species
constituted
by
schemes
satisfying
certain
conditions.
One
then
considers
the
mutation
X
Π
X
that
associates
to
such
a
scheme
X
its
étale
fundamental
group
Π
X
[say,
considered
up
to
inner
automorphisms].
Here,
it
is
important
to
note
that
the
codomain
of
this
mutation
is
the
species
constituted
by
topological
groups
[say,
considered
up
to
inner
automorphisms]
that
satisfy
certain
conditions
which
do
not
include
any
information
concerning
how
the
group
is
related
[for
instance,
via
some
sort
of
étale
fundamental
group
mutation]
to
a
scheme.
The
notion
of
an
anabelian
reconstruction
algorithm
may
then
be
formalized
as
a
mutation
that
forms
a
“mutation-quasi-inverse”
to
the
fundamental
group
mutation.
Another
fundamental
example,
in
the
context
of
the
present
series
of
papers,
of
a
diagram
of
mutations
arises
from
the
Frobenius
morphism
in
positive
characteristic
scheme
theory
[cf.
Example
3.6
for
more
details].
That
is
to
say,
one
fixes
a
prime
number
p
and
considers
the
species
constituted
by
reduced
quasi-compact
schemes
of
characteristic
p
and
quasi-compact
morphisms
of
schemes.
One
then
considers
the
mutation
that
associates
S
S
(p)
to
such
a
scheme
S
the
scheme
S
(p)
with
the
same
topological
space,
but
whose
regular
functions
are
given
by
the
p-th
powers
of
the
regular
functions
on
the
original
scheme.
Thus,
the
domain
and
codomain
of
this
mutation
are
given
by
the
same
species.
One
may
also
consider
a
log
scheme
version
of
this
example,
which,
at
the
level
of
monoids,
corresponds,
in
essence,
to
assigning
M
p
·
M
to
a
torsion-free
abelian
monoid
M
the
submonoid
p
·
M
⊆
M
determined
by
the
image
of
multiplication
by
p.
Returning
to
the
case
of
schemes,
one
may
then
observe
that
the
well-known
constructions
of
the
perfection
and
the
étale
site
S
S
pf
;
S
S
ét
associated
to
a
reduced
scheme
S
of
characteristic
p
give
rise
to
cores
of
the
diagram
obtained
by
considering
iterates
of
the
“Frobenius
mutation”
just
discussed.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
5
This
last
example
of
the
Frobenius
mutation
and
the
associated
core
consti-
tuted
by
the
étale
site
is
of
particular
importance
in
the
context
of
the
present
series
of
papers
in
that
it
forms
the
“intuitive
prototype”
that
underlies
the
theory
of
the
vertical
and
horizontal
lines
of
the
log-theta-lattice
[cf.
the
discussion
of
Remark
3.6.1,
(i)].
One
notable
aspect
of
this
example
is
the
[evident!]
fact
that
the
domain
and
codomain
of
the
Frobenius
mutation
are
given
by
the
same
species.
That
is
to
say,
despite
the
fact
that
in
the
construction
of
the
scheme
S
(p)
[cf.
the
notation
of
the
preceding
paragraph]
from
the
scheme
S,
the
scheme
S
(p)
is
“subordinate”
to
the
scheme
S,
the
domain
and
codomain
species
of
the
resulting
Frobenius
mutation
coincide,
hence,
in
particular,
are
on
a
par
with
one
another.
This
sort
of
situation
served,
for
the
author,
as
a
sort
of
model
for
the
log-
and
Θ
×μ
LGP
-links
of
the
log-theta-lattice,
which
may
be
formulated
as
muta-
tions
between
the
species
constituted
by
the
notion
of
a
Θ
±ell
NF-Hodge
theater.
That
is
to
say,
although
in
the
construction
of
either
the
log-
or
the
Θ
×μ
LGP
-link,
the
±ell
domain
and
codomain
Θ
NF-Hodge
theaters
are
by
no
means
on
a
“par”
with
one
another,
the
domain
and
codomain
Θ
±ell
NF-Hodge
theaters
of
the
resulting
log-/Θ
×μ
LGP
-links
are
regarded
as
objects
of
the
same
species,
hence,
in
particular,
completely
on
a
par
with
one
another.
This
sort
of
“relativization”
of
distinct
models
of
conventional
scheme
theory
over
Z
via
the
notion
of
a
Θ
±ell
NF-Hodge
theater
[cf.
Fig.
I.1
below;
the
discussion
of
“gluing
together”
such
models
of
con-
ventional
scheme
theory
in
[IUTchI],
§I2]
is
one
of
the
most
characteristic
features
of
the
theory
developed
in
the
present
series
of
papers
and,
in
particular,
lies
[tauto-
logically!]
outside
the
framework
of
conventional
scheme
theory
over
Z.
That
is
to
say,
in
the
framework
of
conventional
scheme
theory
over
Z,
if
one
starts
out
with
schemes
over
Z
and
constructs
from
them,
say,
by
means
of
geometric
objects
such
as
the
theta
function
on
a
Tate
curve,
some
sort
of
Frobenioid
that
is
isomorphic
to
a
Frobenioid
associated
to
Z,
then
—
unlike,
for
instance,
the
case
of
the
Frobenius
morphism
in
positive
characteristic
scheme
theory
—
there
is
no
way,
within
the
framework
of
conventional
scheme
theory,
to
treat
the
newly
constructed
Frobenioid
“as
if
it
is
the
Frobenioid
associated
to
Z,
relative
to
some
new
version/model
of
conventional
scheme
theory”.
non-
scheme-
—————
theoretic
link
one
model
of
conven-
tional
scheme
theory
over
Z
non-
scheme-
—————
theoretic
link
another
model
of
conven-
tional
scheme
theory
over
Z
non-
scheme-
—————
theoretic
link
...
...
Fig.
I.1:
Relativized
models
of
conventional
scheme
theory
over
Z
If,
moreover,
one
thinks
of
Z
as
being
constructed,
in
the
usual
way,
via
ax-
iomatic
set
theory,
then
one
may
interpret
the
“absolute”
—
i.e.,
“tautologically
6
SHINICHI
MOCHIZUKI
unrelativizable”
—
nature
of
conventional
scheme
theory
over
Z
at
a
purely
set-
theoretic
level.
Indeed,
from
the
point
of
view
of
the
“∈-structure”
of
axiomatic
set
theory,
there
is
no
way
to
treat
sets
constructed
at
distinct
levels
of
this
∈-structure
as
being
on
a
par
with
one
another.
On
the
other
hand,
if
one
focuses
not
on
the
level
of
the
∈-structure
to
which
a
set
belongs,
but
rather
on
species,
then
the
notion
of
a
species
allows
one
to
relate
—
i.e.,
to
treat
on
a
par
with
one
another
—
objects
belonging
to
the
species
that
arise
from
sets
constructed
at
distinct
levels
of
the
∈-structure.
That
is
to
say,
the
notion
of
a
species
allows
one
to
“simulate
∈-loops”
without
vio-
lating
the
axiom
of
foundation
of
axiomatic
set
theory
—
cf.
the
discussion
of
Remark
3.3.1,
(i).
As
one
constructs
sets
at
new
levels
of
the
∈-structure
of
some
model
of
ax-
iomatic
set
theory
—
e.g.,
as
one
travels
along
vertical
or
horizontal
lines
of
the
log-theta-lattice!
—
one
typically
encounters
new
schemes,
which
give
rise
to
new
Galois
categories,
hence
to
new
Galois
or
étale
fundamental
groups,
which
may
only
be
constructed
if
one
allows
oneself
to
consider
new
basepoints,
relative
to
new
universes.
In
particular,
one
must
continue
to
extend
the
universe,
i.e.,
to
modify
the
model
of
set
theory,
relative
to
which
one
works.
Here,
we
recall
in
passing
that
such
“extensions
of
universe”
are
possible
on
account
of
an
existence
axiom
concerning
universes,
which
is
apparently
attributed
to
the
“Grothendieck
school”
and,
moreover,
cannot,
apparently,
be
obtained
as
a
consequence
of
the
conven-
tional
ZFC
axioms
of
axiomatic
set
theory
[cf.
the
discussion
at
the
beginning
of
§3
for
more
details].
On
the
other
hand,
ultimately
in
the
present
series
of
papers
[cf.
the
discussion
of
[IUTchIII],
Introduction],
we
wish
to
obtain
algorithms
for
constructing
various
objects
that
arise
in
the
context
of
the
new
schemes/universes
discussed
above
—
i.e.,
at
distant
Θ
±ell
NF-Hodge
theaters
of
the
log-theta-lattice
—
that
make
sense
from
the
point
of
view
of
the
original
schemes/universes
that
occurred
at
the
outset
of
the
discussion.
Again,
the
fundamental
tool
that
makes
this
possible,
i.e.,
that
allows
one
to
express
constructions
in
the
new
universes
in
terms
that
makes
sense
in
the
original
universe
is
precisely
the
species-theoretic
formulation
—
i.e.,
the
formulation
via
set-
theoretic
formulas
that
do
not
depend
on
particular
choices
invoked
in
particular
universes
—
of
the
constructions
of
interest
—
cf.
the
discussion
of
Remarks
3.1.2,
3.1.3,
3.1.4,
3.1.5,
3.6.2,
3.6.3.
This
is
the
point
of
view
that
gave
rise
to
the
term
“inter-universal”.
At
a
more
con-
crete
level,
this
“inter-universal”
contact
between
constructions
in
distant
models
of
conventional
scheme
theory
in
the
log-theta-lattice
is
realized
by
considering
[the
étale-like
structures
given
by]
the
various
Galois
or
étale
fundamental
groups
that
occur
as
[the
“type
of
mathematical
object”,
i.e.,
species
constituted
by]
abstract
topological
groups
[cf.
the
discussion
of
Remark
3.6.3,
(i);
[IUTchI],
§I3].
These
abstract
topological
groups
give
rise
to
vertical
or
horizontal
cores
of
the
log-
theta-lattice
[cf.
the
discussion
of
[IUTchIII],
Introduction;
[IUTchIII],
Theorem
1.5,
(i),
(ii)].
Moreover,
once
one
obtains
cores
that
are
sufficiently
“nondegener-
ate”,
or
“rich
in
structure”,
so
as
to
serve
as
containers
for
the
non-coric
portions
of
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
7
the
various
mutations
[e.g.,
vertical
and
horizontal
arrows
of
the
log-theta-lattice]
under
consideration,
then
one
may
construct
the
desired
algorithms,
or
descrip-
tions,
of
these
non-coric
portions
in
terms
of
coric
containers,
up
to
certain
relatively
mild
indeterminacies
[i.e.,
which
reflect
the
non-coric
nature
of
these
non-coric
portions!]
—
cf.
the
illustration
of
this
sort
of
situation
given
in
Fig.
I.2
below;
Remark
3.3.1,
(iii);
Remark
3.6.1,
(ii).
In
the
context
of
the
log-theta-lattice,
this
is
precisely
the
sort
of
situation
that
was
achieved
in
[IUTchIII],
Theorem
A
[cf.
the
discussion
of
[IUTchIII],
Introduction].
...
•
•
•
•
•
•
•
•
...
?
•
Fig.
I.2:
A
coric
container
underlying
a
sequence
of
mutations
In
the
context
of
the
above
discussion
of
set-theoretic
aspects
of
the
theory
developed
in
the
present
series
of
papers,
it
is
of
interest
to
note
the
following
observation,
relative
to
the
analogy
between
the
theory
of
the
present
series
of
papers
and
p-adic
Teichmüller
theory
[cf.
the
discussion
of
[IUTchI],
§I4].
If,
instead
of
working
species-theoretically,
one
attempts
to
document
all
of
the
possible
choices
that
occur
in
various
newly
introduced
universes
that
occur
in
a
construc-
tion,
then
one
finds
that
one
is
obliged
to
work
with
sets,
such
as
sets
obtained
via
set-theoretic
exponentiation,
of
very
large
cardinality.
Such
sets
of
large
cardinality
are
reminiscent
of
the
exponentially
large
denominators
that
occur
if
one
attempts
to
p-adically
formally
integrate
an
arbitrary
connection
as
opposed
to
a
canonical
crystalline
connection
of
the
sort
that
occurs
in
the
context
of
the
canonical
liftings
of
p-adic
Teichmüller
theory
[cf.
the
discussion
of
Remark
3.6.2,
(iii)].
In
this
context,
it
is
of
interest
to
recall
the
computations
of
[Finot],
which
assert,
roughly
speaking,
that
the
canonical
liftings
of
p-adic
Teichmüller
theory
may,
in
certain
cases,
be
characterized
as
liftings
“of
minimal
complexity”
in
the
sense
that
their
Witt
vector
coordinates
are
given
by
polynomials
of
minimal
degree.
Finally,
we
observe
that
although,
in
the
above
discussion,
we
concentrated
on
the
similarities,
from
an
“inter-universal”
point
of
view,
between
the
vertical
and
horizontal
arrows
of
the
log-theta-lattice,
there
is
one
important
difference
between
these
vertical
and
horizontal
arrows:
namely,
·
whereas
the
copies
of
the
full
arithmetic
fundamental
group
—
i.e.,
in
particular,
the
copies
of
the
geometric
fundamental
group
—
on
either
side
of
a
vertical
arrow
are
identified
with
one
another,
·
in
the
case
of
a
horizontal
arrow,
only
the
Galois
groups
of
the
local
base
fields
on
either
side
of
the
arrow
are
identified
with
one
another
8
SHINICHI
MOCHIZUKI
—
cf.
the
discussion
of
Remark
3.6.3,
(ii).
One
way
to
understand
the
reason
for
this
difference
is
as
follows.
In
the
case
of
the
vertical
arrows
—
i.e.,
the
log-
links,
which,
in
essence,
amount
to
the
various
local
p-adic
logarithms
—
in
order
to
construct
the
log-link,
it
is
necessary
to
make
use,
in
an
essential
way,
of
the
local
ring
structures
at
v
∈
V
[cf.
the
discussion
of
[IUTchIII],
Definition
1.1,
(i),
(ii)],
which
may
only
be
reconstructed
from
the
full
arithmetic
fundamental
group.
By
contrast,
in
order
to
construct
the
horizontal
arrows
—
i.e.,
the
Θ
×μ
LGP
-
links
—
this
local
ring
structure
is
unnecessary.
On
the
other
hand,
in
order
to
construct
the
horizontal
arrows,
it
is
necessary
to
work
with
structures
that,
up
to
isomorphism,
are
common
to
both
the
domain
and
the
codomain
of
the
arrow.
Since
the
construction
of
the
domain
of
the
Θ
×μ
LGP
-link
depends,
in
an
essential
way,
on
the
Gaussian
monoids,
i.e.,
on
the
labels
∈
F
l
for
the
theta
values,
which
are
constructed
from
the
geometric
fundamental
group,
while
the
codomain
only
involves
monoids
arising
from
the
local
q-parameters
“q
”
[for
v
∈
V
bad
],
which
v
are
constructed
in
a
fashion
that
is
independent
of
these
labels,
in
order
to
obtain
an
isomorphism
between
structures
arising
from
the
domain
and
codomain,
it
is
necessary
to
restrict
one’s
attention
to
the
Galois
groups
of
the
local
base
fields,
which
are
free
of
any
dependence
on
these
labels.
Acknowledgements:
The
research
discussed
in
the
present
paper
profited
enormously
from
the
gen-
erous
support
that
the
author
received
from
the
Research
Institute
for
Mathematical
Sciences,
a
Joint
Usage/Research
Center
located
in
Kyoto
University.
At
a
personal
level,
I
would
like
to
thank
Fumiharu
Kato,
Akio
Tamagawa,
Go
Yamashita,
Mo-
hamed
Saı̈di,
Yuichiro
Hoshi,
Ivan
Fesenko,
Fucheng
Tan,
Emmanuel
Lepage,
Arata
Minamide,
and
Wojciech
Porowski
for
many
stimulating
discussions
concerning
the
material
presented
in
this
paper.
Also,
I
feel
deeply
indebted
to
Go
Yamashita,
Mohamed
Saı̈di,
and
Yuichiro
Hoshi
for
their
meticulous
reading
of
and
numer-
ous
comments
concerning
the
present
paper.
In
addition,
I
would
like
to
thank
Kentaro
Sato
for
useful
comments
concerning
the
set-theoretic
and
foundational
aspects
of
the
present
paper,
as
well
as
Vesselin
Dimitrov
and
Akshay
Venkatesh
for
useful
comments
concerning
the
analytic
number
theory
aspects
of
the
present
paper.
Finally,
I
would
like
to
express
my
deep
gratitude
to
Ivan
Fesenko
for
his
quite
substantial
efforts
to
disseminate
—
for
instance,
in
the
form
of
a
survey
that
he
wrote
—
the
theory
discussed
in
the
present
series
of
papers.
Notations
and
Conventions:
We
shall
continue
to
use
the
“Notations
and
Conventions”
of
[IUTchI],
§0.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
9
Section
1:
Log-volume
Estimates
In
the
present
§1,
we
perform
various
elementary
local
computations
con-
cerning
nonarchimedean
and
archimedean
local
fields
which
allow
us
to
obtain
more
explicit
versions
[cf.
Theorem
1.10
below]
of
the
log-volume
estimates
for
Θ-
pilot
objects
obtained
in
[IUTchIII],
Corollary
3.12.
In
the
following,
if
λ
∈
R,
then
we
shall
write
λ
(respectively,
λ
)
for
the
smallest
(respectively,
largest)
n
∈
Z
such
that
n
≥
λ
(respectively,
n
≤
λ).
Also,
we
shall
write
“log(−)”
for
the
natural
logarithm
of
a
positive
real
number.
Proposition
1.1.
(Multiple
Tensor
Products
and
Differents)
Let
p
be
a
prime
number,
I
a
finite
set
of
cardinality
≥
2,
Q
p
an
algebraic
closure
of
Q
p
.
×
Write
R
⊆
Q
p
for
the
ring
of
integers
of
Q
p
and
ord
:
Q
p
→
Q
for
the
natural
p-adic
valuation
on
Q
p
,
normalized
so
that
ord(p)
=
1;
for
λ
∈
Q,
we
shall
write
p
λ
for
“some”
[unspecified]
element
of
Q
p
such
that
ord(p
λ
)
=
λ.
For
i
∈
I,
let
def
k
i
⊆
Q
p
be
a
finite
extension
of
Q
p
;
write
R
i
=
O
k
i
=
R
k
i
for
the
ring
of
integers
of
k
i
and
d
i
∈
Q
≥0
for
the
order
[i.e.,
“ord(−)”]
of
any
generator
of
the
different
ideal
of
R
i
over
Z
p
.
Also,
for
any
nonempty
subset
E
⊆
I,
let
us
write
R
E
def
=
R
i
;
d
E
def
=
i∈E
d
i
i∈E
def
—
where
the
tensor
product
is
over
Z
p
.
Fix
an
element
∗
∈
I;
write
I
∗
=
I
\
{∗}.
Then
p
d
I
∗
·
(R
I
)
∼
⊆
R
I
⊆
(R
I
)
∼
—
where
we
write
“(−)
∼
”
for
the
normalization
of
the
[reduced]
ring
in
paren-
theses
in
its
ring
of
fractions,
and
we
observe
that
it
follows
immediately
from
the
definition
of
the
“normalization”
that
the
notation
on
the
left-hand
side
of
the
first
inclusion
of
the
above
display
is
well-defined
for
suitable
“p
d
I
∗
”
[such
as
products
of
elements
p
d
i
∈
R
i
,
for
i
∈
I
∗
]
and
independent
of
the
choice
of
such
suitable
“p
d
I
∗
”.
Proof.
Let
us
regard
R
I
as
an
R
∗
-algebra
in
the
evident
fashion.
It
is
immediate
from
the
definitions
that
R
I
⊆
(R
I
)
∼
.
Now
observe
that
R
⊗
R
∗
R
I
⊆
R
⊗
R
∗
(R
I
)
∼
⊆
(R
⊗
R
∗
R
I
)
∼
—
where
(R
⊗
R
∗
R
I
)
∼
decomposes
as
a
direct
sum
of
finitely
many
copies
of
R.
In
particular,
one
verifies
immediately,
in
light
of
the
fact
the
R
is
faithfully
flat
over
R
∗
,
that
to
complete
the
proof
of
Proposition
1.1,
it
suffices
to
verify
that
p
d
I
∗
·
(R
⊗
R
∗
R
I
)
∼
⊆
R
⊗
R
∗
R
I
10
SHINICHI
MOCHIZUKI
—
where
we
observe
that
it
follows
immediately
from
the
definition
of
the
“nor-
malization”
that
the
notation
on
the
left-hand
side
of
the
inclusion
of
the
above
display
is
well-defined
and
independent
of
the
choice
of
“p
d
I
∗
”.
On
the
other
hand,
it
follows
immediately
from
induction
on
the
cardinality
of
I
that
to
verify
this
last
inclusion,
it
suffices
to
verify
the
inclusion
in
the
case
where
I
is
of
cardinality
two.
But
in
this
case,
the
desired
inclusion
follows
immediately
from
the
definition
of
the
different
ideal.
This
completes
the
proof
of
Proposition
1.1.
Proposition
1.2.
(Differents
and
Logarithms)
We
continue
to
use
the
notation
of
Proposition
1.1.
For
i
∈
I,
write
e
i
for
the
ramification
index
of
k
i
over
Q
p
;
def
a
i
=
1
e
i
if
p
>
2,
·
e
i
p
−
2
def
def
b
i
=
a
i
=
2
if
p
=
2;
1
log(p
·
e
i
/(p
−
1))
−
.
log(p)
e
i
Thus,
1
=
−b
i
.
e
i
if
p
>
2
and
e
i
≤
p
−
2,
then
a
i
=
For
any
nonempty
subset
E
⊆
I,
let
us
write
×
log
p
(R
E
)
def
=
log
p
(R
i
×
);
def
a
E
=
i∈E
a
i
;
i∈E
b
E
def
=
b
i
i∈E
—
where
the
tensor
product
is
over
Z
p
;
we
write
“log
p
(−)”
for
the
p-adic
logarithm.
For
λ
∈
e
1
i
·
Z,
we
shall
write
p
λ
·
R
i
for
the
fractional
ideal
of
R
i
generated
by
any
element
“p
λ
”
of
k
i
such
that
ord(p
λ
)
=
λ.
Let
∼
φ
:
log
p
(R
I
×
)
⊗
Q
p
→
log
p
(R
I
×
)
⊗
Q
p
be
an
automorphism
of
the
finite
dimensional
Q
p
-vector
space
log
p
(R
I
×
)⊗Q
p
that
induces
an
automorphism
of
the
submodule
log
p
(R
I
×
).
Then:
(i)
We
have:
p
a
i
·
R
i
⊆
log
p
(R
i
×
)
⊆
p
−b
i
·
R
i
—
where
the
“⊆’s”
are
equalities
when
p
>
2
and
e
i
≤
p
−
2.
(ii)
We
have:
φ(p
λ
·
(R
I
)
∼
)
⊆
p
λ−d
I
−a
I
·
log
p
(R
I
×
)
⊆
p
λ−d
I
−a
I
−b
I
·
(R
I
)
∼
for
any
λ
∈
e
1
i
·
Z,
i
∈
I.
[Here,
we
observe
that,
just
as
in
Proposition
1.1,
it
follows
immediately
from
the
definition
of
the
“normalization”
that
the
notation
of
the
above
display
is
well-defined
and
independent
of
the
various
choices
involved.]
In
particular,
φ((R
I
)
∼
)
⊆
p
−
d
I
+a
I
·
log
p
(R
I
×
)
⊆
p
−
d
I
+a
I
−b
I
·
(R
I
)
∼
.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
11
(iii)
Suppose
that
p
>
2,
and
that
e
i
≤
p
−
2
for
all
i
∈
I.
Then
we
have:
φ(p
λ
·
(R
I
)
∼
)
⊆
p
λ−d
I
−1
·
(R
I
)
∼
for
any
λ
∈
e
1
i
·
Z,
i
∈
I.
[Here,
we
observe
that,
just
as
in
Proposition
1.1,
it
follows
immediately
from
the
definition
of
the
“normalization”
that
the
notation
of
the
above
display
is
well-defined
and
independent
of
the
various
choices
involved.]
In
particular,
φ((R
I
)
∼
)
⊆
p
−d
I
−1
·
(R
I
)
∼
.
(iv)
If
p
>
2
and
e
i
=
1
for
all
i
∈
I,
then
φ((R
I
)
∼
)
⊆
(R
I
)
∼
.
bi
+
1
ei
1
1
Proof.
Since
a
i
>
p−1
,
p
e
i
>
p−1
[cf.
the
definition
of
“−”,
“−
”!],
asser-
tion
(i)
follows
immediately
from
the
well-known
theory
of
the
p-adic
logarithm
and
exponential
maps
[cf.,
e.g.,
[Kobl],
p.
81].
Next,
we
consider
assertion
(ii).
Observe
that
it
follows
from
the
first
displayed
inclusion
[of
R
I
-modules!]
of
Proposition
1.1
that
⊆
R
I
=
p
a
i
·
R
i
R
i
p
d
I
+a
I
·
(R
I
)
∼
⊆
i∈I
i∈I
and
hence
that
p
λ
·
(R
I
)
∼
⊆
p
λ−d
I
−a
I
·
p
d
I
+a
I
·
(R
I
)
∼
⊆
p
λ−d
I
−a
I
·
p
d
I
+a
I
·
(R
I
)
∼
⊆
p
λ−d
I
−a
I
·
log
p
(R
I
×
)
⊆
p
λ−d
I
−a
I
−b
I
·
(R
I
)
∼
—
where,
in
the
passage
to
the
third
and
fourth
inclusions
following
“p
λ
·(R
I
)
∼
”,
we
apply
assertion
(i).
[Here,
we
observe
that,
just
as
in
Proposition
1.1,
it
follows
im-
mediately
from
the
definition
of
the
“normalization”
that
the
notation
of
the
above
two
displays
is
well-defined
and
independent
of
the
various
choices
involved.]
Thus,
assertion
(ii)
follows
immediately
from
the
fact
that
φ
induces
an
automorphism
of
the
submodule
log
p
(R
I
×
).
Assertion
(iii)
follows
from
assertion
(ii),
together
with
the
fact
that
if
p
>
2
and
e
i
≤
p
−
2
for
all
i
∈
I,
then
we
have
a
I
=
−b
I
,
which
implies
that
λ
−
d
I
−
a
I
−
b
I
≥
λ
−
d
I
−
a
I
−
1
−
b
I
≥
λ
−
d
I
−
1.
Assertion
(iv)
follows
from
assertion
(ii),
together
with
the
fact
that
if
p
>
2
and
e
i
=
1
for
all
i
∈
I,
then
we
have
d
I
=
0,
a
I
=
−b
I
∈
Z.
This
completes
the
proof
of
Proposition
1.2.
Proposition
1.3.
(Estimates
of
Differents)
We
continue
to
use
the
notation
of
Proposition
1.2.
Suppose
that
k
0
⊆
k
i
is
a
subfield
that
contains
Q
p
.
Write
def
R
0
=
O
k
0
for
the
ring
of
integers
of
k
0
,
d
0
for
the
order
[i.e.,
“ord(−)”]
of
any
generator
of
the
different
ideal
of
R
0
over
Z
p
,
e
0
for
the
ramification
index
of
k
0
def
over
Q
p
,
e
i/0
=
e
i
/e
0
(∈
Z),
[k
i
:
k
0
]
for
the
degree
of
the
extension
k
i
/k
0
,
n
i
for
the
unique
nonnegative
integer
such
that
[k
i
:
k
0
]/p
n
i
is
an
integer
prime
to
p.
Then:
(i)
We
have:
d
i
≥
d
0
+
(e
i/0
−
1)/(e
i/0
·
e
0
)
=
d
0
+
(e
i/0
−
1)/e
i
12
SHINICHI
MOCHIZUKI
—
where
the
“≥”
is
an
equality
if
k
i
is
tamely
ramified
over
k
0
.
(ii)
Suppose
that
k
i
is
a
finite
Galois
extension
of
a
subfield
k
1
⊆
k
i
such
that
k
0
⊆
k
1
,
and
k
1
is
tamely
ramified
over
k
0
.
Then
we
have:
d
i
≤
d
0
+
n
i
+
1/e
0
.
Proof.
By
replacing
k
0
by
an
unramified
extension
of
k
0
contained
in
k
i
,
we
may
assume
without
loss
of
generality
in
the
following
discussion
that
k
i
is
a
totally
ramified
extension
of
k
0
.
First,
we
consider
assertion
(i).
Let
π
0
be
a
uniformizer
of
∼
R
0
.
Then
there
exists
an
isomorphism
of
R
0
-algebras
R
0
[x]/(f
(x))
→
R
i
,
where
f
(x)
∈
R
0
[x]
is
a
monic
polynomial
which
is
≡
x
e
i/0
(mod
π
0
),
that
maps
x
→
π
i
for
some
uniformizer
π
i
of
R
i
.
Thus,
the
different
d
i
may
be
computed
as
follows:
e
−1
d
i
−
d
0
=
ord(f
(π
i
))
≥
min(ord(π
0
),
ord(e
i/0
·
π
i
i/0
))
1
1
e
−
1
e
i/0
−
1
e
−1
i/0
≥
min
=
,
ord(π
i
i/0
)
=
min
,
e
0
e
0
e
i/0
·
e
0
e
i
def
—
where,
for
λ,
μ
∈
R
such
that
λ
≥
μ,
we
define
min(λ,
μ)
=
μ.
When
k
i
is
tamely
ramified
over
k
0
,
one
verifies
immediately
that
the
inequalities
of
the
above
display
are,
in
fact,
equalities.
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
We
apply
induction
on
n
i
.
Since
assertion
(ii)
follows
immediately
from
assertion
(i)
when
n
i
=
0,
we
may
assume
that
n
i
≥
1,
and
that
assertion
(ii)
has
been
verified
for
smaller
“n
i
”.
By
replacing
k
1
by
some
tamely
ramified
extension
of
k
1
contained
in
k
i
,
we
may
assume
without
loss
of
generality
that
Gal(k
i
/k
1
)
is
a
p-group.
Since
p-groups
are
solvable,
and
k
i
is
a
totally
ramified
extension
of
k
0
,
it
follows
that
there
exists
a
subextension
k
1
⊆
k
∗
⊆
k
i
such
that
k
i
/k
∗
and
k
∗
/k
1
are
Galois
extensions
of
degree
p
and
p
n
i
−1
,
respectively.
Write
def
R
∗
=
O
k
∗
for
the
ring
of
integers
of
k
∗
,
d
∗
for
the
order
[i.e.,
“ord(−)”]
of
any
generator
of
the
different
ideal
of
R
∗
over
Z
p
,
and
e
∗
for
the
ramification
index
of
k
∗
over
Q
p
.
Thus,
by
the
induction
hypothesis,
it
follows
that
d
∗
≤
d
0
+n
i
−1+1/e
0
.
To
verify
that
d
i
≤
d
0
+
n
i
+
1/e
0
,
it
suffices
to
verify
that
d
i
≤
d
0
+
n
i
+
1/e
0
+
for
any
positive
real
number
.
Thus,
let
us
fix
a
positive
real
number
.
Then
by
possibly
enlarging
k
i
and
k
1
,
we
may
also
assume
without
loss
of
generality
that
the
tamely
ramified
extension
k
1
of
k
0
contains
a
primitive
p-th
root
of
unity,
and,
moreover,
that
the
ramification
index
e
1
of
k
1
over
Q
p
satisfies
the
inequality
e
1
≥
p/
[so
e
∗
≥
e
1
≥
p/].
Thus,
k
i
is
a
Kummer
extension
of
k
∗
.
In
particular,
there
exists
an
inclusion
of
R
∗
-algebras
R
∗
[x]/(f
(x))
→
R
i
,
where
f
(x)
∈
R
∗
[x]
is
a
monic
polynomial
which
is
of
the
form
f
(x)
=
x
p
−
∗
for
some
element
∗
of
R
∗
satisfying
the
estimates
0
≤
ord(
∗
)
≤
p−1
e
∗
,
that
maps
x
→
i
for
some
p−1
element
i
of
R
i
satisfying
the
estimates
0
≤
ord(
i
)
≤
p·e
.
Now
we
compute:
∗
d
i
≤
ord(f
(
i
))
+
d
∗
≤
ord(p
·
i
p−1
)
+
d
0
+
n
i
−
1
+
1/e
0
=
(p
−
1)
·
ord(
i
)
+
d
0
+
n
i
+
1/e
0
≤
≤
(p
−
1)
2
+
d
0
+
n
i
+
1/e
0
p
·
e
∗
p
+
d
0
+
n
i
+
1/e
0
≤
d
0
+
n
i
+
1/e
0
+
e
∗
—
thus
completing
the
proof
of
assertion
(ii).
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
13
Remark
1.3.1.
Similar
estimates
to
those
discussed
in
Proposition
1.3
may
be
found
in
[Ih],
Lemma
A.
Proposition
1.4.
(Nonarchimedean
Normalized
Log-volume
Estimates)
We
continue
to
use
the
notation
of
Proposition
1.2.
Also,
for
i
∈
I,
write
R
i
μ
⊆
R
i
×
for
the
torsion
subgroup
of
R
i
×
,
R
i
×μ
=
R
i
×
/R
i
μ
,
p
f
i
for
the
cardinality
of
the
residue
field
of
k
i
,
and
p
m
i
for
the
order
of
the
p-primary
component
of
R
i
μ
.
Thus,
the
order
of
R
i
μ
is
equal
to
p
m
i
·
(p
f
i
−
1).
Then:
def
(i)
The
log-volumes
constructed
in
[AbsTopIII],
Proposition
5.7,
(i),
on
the
various
finite
extensions
of
Q
p
contained
in
Q
p
may
be
suitably
normalized
[i.e.,
by
dividing
by
the
degree
of
the
finite
extension]
so
as
to
yield
a
notion
of
log-volume
μ
log
(−)
defined
on
compact
open
subsets
of
finite
extensions
of
Q
p
contained
in
Q
p
,
valued
in
R,
and
normalized
so
that
μ
log
(R
i
)
=
0,
μ
log
(p
·
R
i
)
=
−log(p),
for
each
i
∈
I.
Moreover,
by
applying
the
fact
that
tensor
products
of
finitely
many
finite
extensions
of
Q
p
over
Z
p
decompose,
naturally,
as
direct
sums
of
finitely
many
finite
extensions
of
Q
p
,
we
obtain
a
notion
of
log-volume
—
which,
by
abuse
of
notation,
we
shall
also
denote
by
“μ
log
(−)”
—
defined
on
compact
open
subsets
of
such
tensor
products,
valued
in
R,
and
normalized
so
that
μ
log
((R
E
)
∼
)
=
0,
μ
log
(p
·
(R
E
)
∼
)
=
−log(p),
for
any
nonempty
set
E
⊆
I.
(ii)
We
have:
μ
log
(log
p
(R
i
×
))
=
−
1
e
i
+
m
i
·
log(p)
e
i
f
i
[cf.
[AbsTopIII],
Proposition
5.8,
(iii)].
(iii)
Let
I
∗
⊆
I
be
a
subset
such
that
for
each
i
∈
I
\
I
∗
,
it
holds
that
p
−
2
≥
e
i
(≥
1).
Then
for
any
λ
∈
e
1
†
·
Z,
i
†
∈
I,
we
have
inclusions
φ(p
λ
·
(R
I
)
∼
)
⊆
i
p
λ−d
I
−a
I
·
log
p
(R
I
×
)
⊆
p
λ−d
I
−a
I
−b
I
·
(R
I
)
∼
and
inequalities
μ
log
λ−d
I
−a
I
(p
·
log
p
(R
I
×
))
μ
log
(p
λ−d
I
−a
I
−b
I
·
(R
I
)
∼
)
≤
≤
−
λ
+
d
I
+
1
+
4
·
|I
|/p
·
log(p);
∗
−
λ
+
d
I
+
1
·
log(p)
+
{3
+
log(e
i
)}
i∈I
∗
—
where
we
write
“|(−)|”
for
the
cardinality
of
the
set
“(−)”.
Moreover,
d
I
+
a
I
≥
|I|
if
p
>
2;
d
I
+
a
I
≥
2
·
|I|
if
p
=
2.
μ
log
(iv)
If
p
>
2
and
e
i
=
1
for
all
i
∈
I,
then
φ((R
I
)
∼
)
((R
I
)
∼
)
=
0.
⊆
(R
I
)
∼
,
and
Proof.
Assertion
(i)
follows
immediately
from
the
definitions.
Next,
we
consider
assertion
(ii).
We
begin
by
observing
that
every
compact
open
subset
of
R
i
×μ
may
be
covered
by
a
finite
collection
of
compact
open
subsets
of
R
i
×μ
that
arise
as
14
SHINICHI
MOCHIZUKI
images
of
compact
open
subsets
of
R
i
×
that
map
injectively
to
R
i
×μ
.
In
particular,
by
applying
this
observation,
we
conclude
that
the
log-volume
on
R
i
×
determines,
in
a
natural
way,
a
log-volume
on
the
quotient
R
i
×
R
i
×μ
.
Moreover,
in
light
of
the
compatibility
of
the
log-volume
with
“log
p
(−)”
[cf.
[AbsTopIII],
Proposition
5.7,
(i),
(c)],
it
follows
immediately
that
μ
log
(log
p
(R
i
×
))
=
μ
log
(R
i
×μ
).
Thus,
it
suffices
to
compute
e
i
·
f
i
·
μ
log
(R
i
×μ
)
=
e
i
·
f
i
·
μ
log
(R
i
×
)
−
log(p
m
i
·
(p
f
i
−
1)).
On
the
other
hand,
it
follows
immediately
from
the
basic
properties
of
the
log-volume
[cf.
[AbsTopIII],
Proposition
5.7,
(i),
(a)]
that
e
i
·
f
i
·
μ
log
(R
i
×
)
=
log(1
−
p
−f
i
),
so
e
i
·
f
i
·
μ
log
(R
i
×μ
)
=
−(f
i
+
m
i
)
·
log(p),
as
desired.
This
completes
the
proof
of
assertion
(ii).
The
inclusions
of
assertion
(iii)
follow
immediately
from
Proposition
1.2,
(ii).
When
p
=
2,
the
fact
that
d
I
+
a
I
≥
2
·
|I|
follows
immediately
from
the
definition
of
“d
i
”
and
“a
i
”
in
Propositions
1.1,
1.2.
When
p
>
2,
it
follows
immediately
from
the
definition
of
“a
i
”
in
Proposition
1.2
that
a
i
≥
1/e
i
,
for
all
i
∈
I;
thus,
since
d
i
≥
(e
i
−
1)/e
i
for
all
i
∈
I
[cf.
Proposition
1.3,
(i)],
we
conclude
that
d
i
+
a
i
≥
1
for
all
i
∈
I,
and
hence
that
d
I
+
a
I
≥
|I|,
as
asserted
in
the
statement
of
p
1
≤
p
4
for
p
≥
3;
p−1
≤
2
for
p
≥
2;
assertion
(iii).
Next,
let
us
observe
that
p−2
2
1
p
≤
log(p)
for
p
≥
2.
Thus,
it
follows
immediately
from
the
definition
of
a
i
,
b
i
in
2
Proposition
1.2
that
a
i
−
e
1
i
≤
p
4
≤
log(p)
,
(b
i
+
e
1
i
)
·
log(p)
≤
log(2e
i
)
≤
1
+
log(e
i
)
for
i
∈
I;
a
i
=
e
1
i
=
−b
i
for
i
∈
I
\
I
∗
.
On
the
other
hand,
by
assertion
(i),
we
have
μ
log
(R
I
)
≤
μ
log
((R
I
)
∼
)
=
0;
by
assertion
(ii),
we
have
μ
log
(log
p
(R
i
×
))
≤
−
e
1
i
·log(p).
Now
we
compute:
μ
log
(p
λ−d
I
−a
I
·
log
p
(R
I
×
))
≤
−
λ
+
d
I
+
a
I
+
1
·
log(p)
+
μ
log
(log
p
(R
I
×
))
=
−
λ
+
d
I
+
a
I
+
1
·
log(p)
μ
log
(log
p
(R
i
×
))
+
μ
log
(R
I
)
+
i∈I
1
(a
i
−
)
·
log(p)
≤
−
λ
+
d
I
+
1
+
e
i
i∈I
≤
−
λ
+
d
I
+
1
+
4
·
|I
∗
|/p
·
log(p);
μ
log
(p
λ−d
I
−a
I
−b
I
·
(R
I
)
∼
)
≤
−
λ
+
d
I
+
a
I
+
b
I
+
1
·
log(p)
{3
+
log(e
i
)}
≤
−
λ
+
d
I
+
1
·
log(p)
+
i∈I
∗
—
thus
completing
the
proof
of
assertion
(iii).
Assertion
(iv)
follows
immediately
from
assertion
(i)
and
Proposition
1.2,
(iv).
Proposition
1.5.
(Archimedean
Metric
Estimates)
In
the
following,
we
shall
regard
the
complex
archimedean
field
C
as
being
equipped
with
its
standard
Hermitian
metric,
i.e.,
the
metric
determined
by
the
complex
norm.
Let
us
refer
to
as
the
primitive
automorphisms
of
C
the
group
of
automorphisms
[of
order
8]
of
the
underlying
metrized
real
vector
space
of
C
generated
by
the
operations
of
√
complex
conjugation
and
multiplication
by
±1
or
±
−1.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
15
(i)
(Direct
Sum
vs.
Tensor
Product
Metrics)
The
metric
on
C
deter-
mines
a
tensor
product
metric
on
C
⊗
R
C,
as
well
as
a
direct
sum
metric
on
C
⊕
C.
Then,
relative
to
these
metrics,
any
isomorphism
of
topological
rings
[i.e.,
arising
from
the
Chinese
remainder
theorem]
∼
C
⊗
R
C
→
C
⊕
C
is
compatible
with
these
metrics,
up
to
a
factor
of
2,
i.e.,
the
metric
on
the
right-
hand
side
corresponds
√
to
2
times
the
metric
on
the
left-hand
side.
[Thus,
lengths
differ
by
a
factor
of
2.]
(ii)
(Direct
Sum
vs.
Tensor
Product
Automorphisms)
Relative
to
the
notation
of
(i),
the
direct
sum
decomposition
C
⊕
C,
together
with
its
Her-
mitian
metric,
is
preserved,
relative
to
the
displayed
isomorphism
of
(i),
by
the
automorphisms
of
C
⊗
R
C
induced
by
the
various
primitive
automorphisms
of
the
two
copies
of
“C”
that
appear
in
the
tensor
product
C
⊗
R
C.
(iii)
(Direct
Sums
and
Tensor
Products
of
Multiple
Copies)
Let
I,
V
be
nonempty
finite
sets,
whose
cardinalities
we
denote
by
|I|,
|V
|,
respectively.
Write
def
=
C
v
M
v∈V
def
for
the
direct
sum
of
copies
C
v
=
C
of
C
labeled
by
v
∈
V
,
which
we
regard
as
equipped
with
the
direct
sum
metric,
and
def
=
M
I
M
i
i∈I
def
for
the
tensor
product
over
R
of
copies
M
i
=
M
of
M
labeled
by
i
∈
I,
which
we
regard
as
equipped
with
the
tensor
product
metric
[cf.
the
constructions
of
[IUTchIII],
Proposition
3.2,
(ii)].
Then
the
topological
ring
structure
on
each
C
v
determines
a
topological
ring
structure
on
M
I
with
respect
to
which
M
I
admits
a
unique
direct
sum
decomposition
as
a
direct
sum
of
2
|I|−1
·
|V
|
|I|
copies
of
C
[cf.
[IUTchIII],
Proposition
3.1,
(i)].
The
direct
sum
metric
on
M
I
—
i.e.,
the
metric
determined
by
the
natural
metrics
on
these
copies
of
C
—
is
equal
to
2
|I|−1
times
the
original
tensor
product
metric
on
M
I
.
Write
B
I
⊆
M
I
for
the
“integral
structure”
[cf.
the
constructions
of
[IUTchIII],
Proposition
3.1,
(ii)]
given
by
the
direct
product
of
the
unit
balls
of
the
copies
of
C
that
occur
in
the
direct
sum
decomposition
of
M
I
.
Then
the
tensor
product
metric
on
M
I
,
the
direct
sum
decomposition
of
M
I
,
the
direct
sum
metric
on
M
I
,
and
the
integral
16
SHINICHI
MOCHIZUKI
structure
B
I
⊆
M
I
are
preserved
by
the
automorphisms
of
M
I
induced
by
the
various
primitive
automorphisms
of
the
direct
summands
“C
v
”
that
appear
in
the
factors
“M
i
”
of
the
tensor
product
M
I
.
(iv)
(Tensor
Product
of
Vectors
of
a
Given
Length)
Suppose
that
we
are
in
the
situation
of
(iii).
Fix
λ
∈
R
>0
.
Then
M
I
m
i
∈
λ
|I|
·
B
I
i∈I
for
any
collection
of
elements
{m
i
∈
M
i
}
i∈I
such
that
the
component
of
m
i
in
each
direct
summand
“C
v
”
of
M
i
is
of
length
λ.
Proof.
Assertions
(i)
and
(ii)
are
discussed
in
[IUTchIII],
Remark
3.9.1,
(ii),
and
may
be
verified
by
means
of
routine
and
elementary
arguments.
Assertion
(iii)
follows
immediately
from
assertions
(i)
and
(ii).
Assertion
(iv)
follows
immediately
from
the
various
definitions
involved.
Proposition
1.6.
(The
Prime
Number
Theorem)
If
n
is
a
positive
integer,
then
let
us
write
p
n
for
the
n-th
smallest
prime
number.
[Thus,
p
1
=
2,
p
2
=
3,
and
so
on.]
Then
there
exists
an
integer
n
0
such
that
it
holds
that
4p
n
3·log(p
n
)
n
≤
for
all
n
≥
n
0
.
In
particular,
there
exists
a
positive
real
number
η
prm
such
that
4η
1
≤
3·log(η)
p≤η
—
where
the
sum
ranges
over
the
prime
numbers
p
≤
η
—
for
all
positive
real
η
≥
η
prm
.
Proof.
Relative
to
our
notation,
the
Prime
Number
Theorem
[cf.,
e.g.,
[DmMn],
§3.10]
implies
that
n
·
log(p
n
)
lim
=
1
n→∞
p
n
—
i.e.,
in
particular,
that
for
some
positive
integer
n
0
,
it
holds
that
log(p
n
)
4
1
·
≤
p
n
3
n
for
all
n
≥
n
0
.
The
final
portion
of
Proposition
1.6
follows
formally.
Proposition
1.7.
(Weighted
Averages)
Let
E
be
a
nonempty
finite
set,
n
a
positive
integer.
For
e
∈
E,
let
λ
e
∈
R
>0
,
β
e
∈
R.
Then,
for
any
i
=
1,
.
.
.
,
n,
we
have:
β
e
·
λ
Πe
n
·
β
e
i
·
λ
Πe
e
∈E
n
e
∈E
n
=
λ
Πe
e
∈E
n
e
∈E
n
=
λ
Πe
n
·
β
avg
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
—
where
we
write
def
β
avg
=
β
E
/λ
E
,
β
e
def
=
n
def
β
E
=
β
e
j
;
e∈E
λ
Πe
17
def
β
e
·
λ
e
,
λ
E
=
e∈E
λ
e
,
n
def
=
λ
e
j
j=1
j=1
for
any
n-tuple
e
=
(e
1
,
.
.
.
,
e
n
)
∈
E
n
of
elements
of
E.
Proof.
We
begin
by
observing
that
λ
nE
=
λ
Πe
;
β
E
·
λ
n−1
E
e
∈E
n
=
β
e
i
·
λ
Πe
e
∈E
n
for
any
i
=
1,
.
.
.
,
n.
Thus,
summing
over
i,
we
obtain
that
n
·
β
E
·
λ
n−1
=
β
·
λ
=
n
·
β
e
i
·
λ
Πe
e
Π
e
E
e
∈E
n
e
∈E
n
and
hence
that
n
n
·
β
avg
=
n
·
β
E
·
λ
n−1
E
/λ
E
=
e
∈E
n
=
β
e
·
λ
Πe
·
λ
Πe
−1
e
∈E
n
n
·
β
e
i
·
λ
Πe
·
e
∈E
n
e
∈E
n
λ
Πe
−1
as
desired.
Remark
1.7.1.
In
Theorem
1.10
below,
we
shall
apply
Proposition
1.7
to
com-
pute
various
packet-normalized
log-volumes
of
the
sort
discussed
in
[IUTchIII],
Proposition
3.9,
(i)
—
i.e.,
log-volumes
normalized
by
means
of
the
normalized
weights
discussed
in
[IUTchIII],
Remark
3.1.1,
(ii).
Here,
we
recall
that
the
nor-
malized
weights
discussed
in
[IUTchIII],
Remark
3.1.1,
(ii),
were
computed
relative
to
the
non-normalized
log-volumes
of
[AbsTopIII],
Proposition
5.8,
(iii),
(vi)
[cf.
the
discussion
of
[IUTchIII],
Remark
3.1.1,
(ii);
[IUTchI],
Example
3.5,
(iii)].
By
contrast,
in
the
discussion
of
the
present
§1,
our
computations
are
performed
rela-
tive
to
normalized
log-volumes
as
discussed
in
Proposition
1.4,
(i).
In
particular,
it
follows
that
the
weights
[K
v
:
(F
mod
)
v
]
−1
,
where
V
v
|
v
∈
V
mod
,
of
the
dis-
cussion
of
[IUTchIII],
Remark
3.1.1,
(ii),
must
be
replaced
—
i.e.,
when
one
works
with
normalized
log-volumes
as
in
Proposition
1.4,
(i)
—
by
the
weights
[K
v
:
Q
v
Q
]
·
[K
v
:
(F
mod
)
v
]
−1
=
[(F
mod
)
v
:
Q
v
Q
]
—
where
V
mod
v
|
v
Q
∈
V
Q
.
This
means
that
the
normalized
weights
of
the
final
display
of
[IUTchIII],
Remark
3.1.1,
(ii),
must
be
replaced,
when
one
works
with
normalized
log-volumes
as
in
Proposition
1.4,
(i),
by
the
normalized
weights
[(F
mod
)
v
α
:
Q
v
Q
]
α∈A
{w
α
}
α∈A
[(F
mod
)
w
α
:
Q
v
Q
]
α∈A
18
SHINICHI
MOCHIZUKI
—
where
the
sum
is
over
all
collections
{w
α
}
α∈A
of
[not
necessarily
distinct!]
ele-
ments
w
α
∈
V
mod
lying
over
v
Q
and
indexed
by
α
∈
A.
Thus,
in
summary,
when
one
works
with
normalized
log-volumes
as
in
Proposition
1.4,
(i),
the
appropriate
normalized
weights
are
given
by
the
expressions
λ
†
Πe
λ
Πe
e
∈E
n
[where
e
†
∈
E
n
]
that
appear
in
Proposition
1.7.
Here,
one
takes
“E”
to
be
the
set
∼
of
elements
of
V
→
V
mod
lying
over
a
fixed
v
Q
;
one
takes
“n”
to
be
the
cardinality
of
A,
so
that
one
can
write
A
=
{α
1
,
.
.
.
,
α
n
}
[where
the
α
i
are
distinct];
if
e
∈
E
corresponds
to
v
∈
V,
v
∈
V
mod
,
then
one
takes
def
“λ
e
”
=
[(F
mod
)
v
:
Q
v
Q
]
∈
R
>0
and
“β
e
”
to
be
a
normalized
log-volume
of
some
compact
open
subset
of
K
v
.
Before
proceeding,
we
review
some
well-known
elementary
facts
concerning
elliptic
curves.
In
the
following,
we
shall
write
M
ell
for
the
moduli
stack
of
elliptic
curves
over
Z
and
M
ell
⊆
M
ell
for
the
natural
compactification
of
M
ell
,
i.e.,
the
moduli
stack
of
one-dimensional
def
semi-abelian
schemes
over
Z.
Also,
if
R
is
a
Z-algebra,
then
we
shall
write
(M
ell
)
R
=
def
M
ell
×
Z
R,
(M
ell
)
R
=
M
ell
×
Z
R.
Proposition
1.8.
(Torsion
Points
of
Elliptic
Curves)
Let
k
be
a
perfect
def
field,
k
an
algebraic
closure
of
k.
Write
G
k
=
Gal(k/k).
(i)
(“Serre’s
Criterion”)
Let
l
≥
3
be
a
prime
number
that
is
invertible
in
k;
suppose
that
k
=
k.
Let
A
be
an
abelian
variety
over
k,
equipped
with
a
polarization
λ.
Write
A[l]
⊆
A(k)
for
the
group
of
l-torsion
points
of
A(k).
Then
the
natural
map
φ
:
Aut
k
(A,
λ)
→
Aut(A[l])
from
the
group
of
automorphisms
of
the
polarized
abelian
variety
(A,
λ)
over
k
to
the
group
of
automorphisms
of
the
abelian
group
A[l]
is
injective.
(ii)
Let
E
k
be
an
elliptic
curve
over
k
with
origin
E
∈
E(k).
For
n
a
positive
integer,
write
E
k
[n]
⊆
E
k
(k)
for
the
module
of
n-torsion
points
of
E
k
(k)
and
Aut
k
(E
k
)
⊆
Aut
k
(E
k
)
for
the
respective
groups
of
E
-preserving
automorphisms
of
the
k-scheme
E
k
and
the
k-scheme
E
k
.
Then
we
have
a
natural
exact
sequence
1
−→
Aut
k
(E
k
)
−→
Aut
k
(E
k
)
−→
G
k
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
19
—
where
the
image
G
E
⊆
G
k
of
the
homomorphism
Aut
k
(E
k
)
→
G
k
is
open
—
and
a
natural
representation
ρ
n
:
Aut
k
(E
k
)
→
Aut(E
k
[n])
on
the
n-torsion
points
of
E
k
.
The
finite
extension
k
E
of
k
determined
by
G
E
is
the
minimal
field
of
definition
of
E
k
,
i.e.,
the
field
generated
over
k
by
the
j-
invariant
of
E
k
.
Finally,
if
H
⊆
G
k
is
any
closed
subgroup,
which
corresponds
to
an
extension
k
H
of
k,
then
the
datum
of
a
model
of
E
k
over
k
H
[i.e.,
descent
data
for
E
k
from
k
to
k
H
]
is
equivalent
to
the
datum
of
a
section
of
the
homomorphism
Aut
k
(E
k
)
→
G
k
over
H.
In
particular,
the
homomorphism
Aut
k
(E
k
)
→
G
k
admits
a
section
over
G
E
.
(iii)
In
the
situation
of
(ii),
suppose
further
that
Aut
k
(E
k
)
=
{±1}.
Then
the
representation
ρ
2
factors
through
G
E
and
hence
defines
a
natural
representa-
tion
G
E
→
Aut(E
k
[2]).
(iv)
In
the
situation
of
(ii),
suppose
further
that
l
≥
3
is
a
prime
number
that
is
invertible
in
k,
and
that
E
k
descends
to
elliptic
curves
E
k
and
E
k
over
k,
all
of
whose
l-torsion
points
are
rational
over
k.
Then
E
k
is
isomorphic
to
E
k
over
k.
(v)
In
the
situation
of
(ii),
suppose
further
that
k
is
a
complete
discrete
valuation
field
with
ring
of
integers
O
k
,
that
l
≥
3
is
a
prime
number
that
is
invertible
in
O
k
,
and
that
E
k
descends
to
an
elliptic
curve
E
k
over
k,
all
of
whose
l-torsion
points
are
rational
over
k.
Then
E
k
has
semi-stable
reduction
over
O
k
[i.e.,
extends
to
a
semi-abelian
scheme
over
O
k
].
(vi)
In
the
situation
of
(iii),
suppose
further
that
2
is
invertible
in
k,
that
G
E
=
G
k
,
and
that
the
representation
G
E
→
Aut(E
k
[2])
is
trivial.
Then
E
k
descends
to
an
elliptic
curve
E
k
over
k
which
is
defined
by
means
of
the
Legendre
form
of
the
Weierstrass
equation
[cf.,
e.g.,
the
statement
of
Corollary
2.2,
below].
If,
moreover,
k
is
a
complete
discrete
valuation
field
with
ring
of
integers
O
k
such
that
2
is
invertible
in
O
k
,
then
E
k
has
semi-stable
reduction
over
O
k
[i.e.,
extends
to
a
semi-abelian
scheme
over
O
k
]
for
some
finite
extension
k
⊆
k
of
k
such
that
[k
:
k]
≤
2;
if
E
k
has
good
reduction
over
O
k
[i.e.,
extends
to
an
abelian
scheme
over
O
k
],
then
one
may
in
fact
take
k
to
be
k.
(vii)
In
the
situation
of
(ii),
suppose
further
that
k
is
a
complete
discrete
valuation
field
with
ring
of
integers
O
k
,
that
E
k
descends
to
an
elliptic
curve
E
k
over
k,
and
that
n
is
invertible
in
O
k
.
If
E
k
has
good
reduction
over
O
k
[i.e.,
extends
to
an
abelian
scheme
over
O
k
],
then
the
action
of
G
k
on
E
k
[n]
is
unramified.
If
E
k
has
bad
multiplicative
reduction
over
O
k
[i.e.,
extends
to
a
non-proper
semi-abelian
scheme
over
O
k
],
then
the
kernel
of
the
action
of
G
k
on
E
k
[n]
determines
a
tamely
ramified
extension
of
k
whose
ramification
index
over
k
divides
n.
Proof.
First,
we
consider
assertion
(i).
Suppose
that
φ
is
not
injective.
Since
Aut
k
(A,
λ)
is
well-known
to
be
finite
[cf.,
e.g.,
[Milne],
Proposition
17.5,
(a)],
we
thus
conclude
that
there
exists
an
α
∈
Ker(φ)
of
order
n
=
1.
We
may
assume
20
SHINICHI
MOCHIZUKI
without
loss
of
generality
that
n
is
prime.
Now
we
follow
the
argument
of
[Milne],
Proposition
17.5,
(b).
Since
α
acts
trivially
on
A[l],
it
follows
immediately
that
the
endomorphism
of
A
given
by
α
−id
A
[where
id
A
denotes
the
identity
automorphism
of
A]
may
be
written
in
the
form
l
·
β,
for
β
an
endomorphism
of
A
over
k.
Write
T
l
(A)
for
the
l-adic
Tate
module
of
A.
Since
α
n
=
id
A
,
it
follows
that
the
eigen-
values
of
the
action
of
α
on
T
l
(A)
are
n-th
roots
of
unity.
On
the
other
hand,
the
eigenvalues
of
the
action
of
β
on
T
l
(A)
are
algebraic
integers
[cf.
[Milne],
Theorem
12.5].
We
thus
conclude
that
each
eigenvalue
ζ
of
the
action
of
α
on
T
l
(A)
is
an
n-th
root
of
unity
which,
as
an
algebraic
integer,
is
≡
1
(mod
l)
[where
l
≥
3],
hence
=
1.
Since
α
n
=
id
A
,
it
follows
that
α
acts
on
T
l
(A)
as
a
semi-simple
matrix
which
is
also
unipotent,
hence
equal
to
the
identity
matrix.
But
this
implies
that
α
=
id
A
[cf.
[Milne],
Theorem
12.5].
This
contradiction
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
Since
E
k
is
proper
over
k,
it
follows
[by
considering
the
space
of
global
sections
of
the
structure
sheaf
of
E
k
]
that
any
automorphism
of
the
scheme
E
k
lies
over
an
automorphism
of
k.
This
implies
the
existence
of
a
natural
exact
sequence
and
natural
representation
as
in
the
statement
of
assertion
(ii).
The
relationship
between
k
E
and
the
j-invariant
of
E
k
follows
immediately
from
the
well-known
theory
of
the
j-invariant
of
an
elliptic
curve
[cf.,
e.g.,
[Silv],
Chapter
III,
Proposition
1.4,
(b),
(c)].
The
final
portion
of
assertion
(ii)
concerning
models
of
E
k
follows
immediately
from
the
definitions.
This
completes
the
proof
of
assertion
(ii).
Assertion
(iii)
follows
immediately
from
the
fact
that
{±1}
acts
trivially
on
E
k
[2].
Next,
we
consider
assertion
(iv).
First,
let
us
observe
that
it
follows
immedi-
ately
from
the
final
portion
of
assertion
(ii)
that
a
model
E
k
∗
of
E
k
over
k
all
of
whose
l-torsion
points
are
rational
over
k
corresponds
to
a
closed
subgroup
H
∗
⊆
Aut
k
(E
k
)
that
lies
in
the
kernel
of
ρ
l
and,
moreover,
maps
isomorphically
to
G
k
.
On
the
other
hand,
it
follows
from
assertion
(i)
that
the
restriction
of
ρ
l
to
Aut
k
(E
k
)
⊆
Aut
k
(E
k
)
is
injective.
Thus,
the
closed
subgroup
H
∗
⊆
Aut
k
(E
k
)
is
uniquely
determined
by
the
condition
that
it
lie
in
the
kernel
of
ρ
l
and,
moreover,
map
isomorphically
to
G
k
.
This
completes
the
proof
of
assertion
(iv).
Next,
we
consider
assertion
(v).
First,
let
us
observe
that,
by
considering
l-
level
structures,
we
obtain
a
finite
covering
of
S
→
(M
ell
)
Z[
1
l
]
which
is
étale
over
(M
ell
)
Z[
1
l
]
and
tamely
ramified
over
the
divisor
at
infinity.
Then
it
follows
from
assertion
(i)
that
the
algebraic
stack
S
is
in
fact
a
scheme,
which
is,
moreover,
proper
over
Z[
1
l
].
Thus,
it
follows
from
the
valuative
criterion
for
properness
that
any
k-valued
point
of
S
determined
by
E
k
—
where
we
observe
that
such
a
point
necessarily
exists,
in
light
of
our
assumption
that
the
l-torsion
points
of
E
k
are
rational
over
k
—
extends
to
an
O
k
-valued
point
of
S,
hence
also
of
M
ell
,
as
desired.
This
completes
the
proof
of
assertion
(v).
Next,
we
consider
assertion
(vi).
Since
G
E
=
G
k
,
it
follows
from
assertion
(ii)
that
E
k
descends
to
an
elliptic
curve
E
k
over
k.
Our
assumption
that
the
representation
G
k
=
G
E
→
Aut(E
k
[2])
of
assertion
(iii)
is
trivial
implies
that
the
2-torsion
points
of
E
k
are
rational
over
k.
Thus,
by
considering
suitable
global
sections
of
tensor
powers
of
the
line
bundle
on
E
k
determined
by
the
origin
on
which
the
automorphism
“−1”
of
E
k
acts
via
multiplication
by
±1
[cf.,
e.g.,
[Harts],
Chapter
IV,
the
proof
of
Proposition
4.6],
one
concludes
immediately
that
a
suitable
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
21
[possibly
trivial]
twist
E
k
of
E
k
over
k
[i.e.,
such
that
E
k
and
E
k
are
isomorphic
over
some
quadratic
extension
k
of
k]
may
be
defined
by
means
of
the
Legendre
form
of
the
Weierstrass
equation.
Now
suppose
that
k
is
a
complete
discrete
valuation
field
with
ring
of
integers
O
k
such
that
2
is
invertible
in
O
k
,
and
that
E
k
is
defined
by
means
of
the
Legendre
form
of
the
Weierstrass
equation.
Then
the
fact
that
E
k
has
semi-stable
reduction
over
O
k
for
some
finite
extension
k
⊆
k
of
k
such
that
[k
:
k]
≤
2
follows
from
the
explicit
computations
of
the
proof
of
[Silv],
Chapter
VII,
Proposition
5.4,
(c).
These
explicit
computations
also
imply
that
if
E
k
has
good
reduction
over
O
k
,
then
one
may
in
fact
take
k
to
be
k.
This
completes
the
proof
of
assertion
(vi).
Assertion
(vii)
follows
immediately
from
[NerMod],
§7.4,
Theorem
5,
in
the
case
of
good
reduction
and
from
[NerMod],
§7.4,
Theorem
6,
in
the
case
of
bad
multiplicative
reduction.
We
are
now
ready
to
apply
the
elementary
computations
discussed
above
to
give
more
explicit
log-volume
estimates
for
Θ-pilot
objects.
We
begin
by
recalling
some
notation
and
terminology
from
[GenEll],
§1.
Definition
1.9.
Let
F
be
a
number
field
[i.e.,
a
finite
extension
of
the
ra-
tional
number
field
Q],
whose
set
of
valuations
we
denote
by
V(F
).
Thus,
V(F
)
V(F
)
arc
of
nonarchimedean
decomposes
as
a
disjoint
union
V(F
)
=
V(F
)
non
and
archimedean
valuations.
If
v
∈
V(F
),
then
we
shall
write
F
v
for
the
completion
of
F
at
v;
if
v
∈
V(F
)
non
,
then
we
shall
write
e
v
for
the
ramification
index
of
F
v
over
Q
p
v
,
f
v
for
the
residue
field
degree
of
F
v
over
Q
p
v
,
and
q
v
for
the
cardinality
of
the
residue
field
of
F
v
.
(i)
An
[R-]arithmetic
divisor
a
on
F
is
defined
to
be
a
finite
formal
sum
c
v
·
v
v∈V(F
)
—
where
c
v
∈
R,
for
all
v
∈
V(F
).
Here,
we
shall
refer
to
the
set
Supp(a)
of
v
∈
V(F
)
such
that
c
v
=
0
as
the
support
of
a;
if
all
of
the
c
v
are
≥
0,
then
we
shall
say
that
the
arithmetic
divisor
is
effective.
Thus,
the
[R-]arithmetic
divisors
on
F
naturally
form
a
group
ADiv
R
(F
).
The
assignment
V(F
)
non
v
→
log(q
v
);
V(F
)
arc
v
→
1
determines
a
homomorphism
deg
F
:
ADiv
R
(F
)
→
R
which
we
shall
refer
to
as
the
degree
map.
If
a
∈
ADiv
R
(F
),
then
we
shall
refer
to
deg(a)
def
=
1
·
deg
F
(a)
[F
:
Q]
22
SHINICHI
MOCHIZUKI
as
the
normalized
degree
of
a.
Thus,
for
any
finite
extension
K
of
F
,
we
have
deg(a|
K
)
=
deg(a)
—
where
we
write
deg(a|
K
)
for
the
normalized
degree
of
the
pull-back
a|
K
∈
ADiv
R
(K)
[defined
in
the
evident
fashion]
of
a
to
K.
def
(ii)
Let
v
Q
∈
V
Q
=
V(Q),
E
⊆
V(F
)
a
nonempty
set
of
elements
lying
over
v
Q
.
If
a
=
c
v
·
v
∈
ADiv
R
(F
),
then
we
shall
write
v∈V(F
)
def
a
E
=
c
v
·
v
∈
ADiv
R
(F
);
v∈E
deg(a
E
)
def
deg
E
(a)
=
[F
v
:
Q
v
Q
]
v∈E
for
the
portion
of
a
supported
in
E
and
the
“normalized
E-degree”
of
a,
respectively.
Thus,
for
any
finite
extension
K
of
F
,
we
have
deg
E|
K
(a|
K
)
=
deg
E
(a)
—
where
we
write
E|
K
⊆
V(K)
for
the
set
of
valuations
lying
over
valuations
∈
E.
Theorem
1.10.
(Log-volume
Estimates
for
Θ-Pilot
Objects)
Fix
a
col-
lection
of
initial
Θ-data
as
in
[IUTchI],
Definition
3.1.
Suppose
that
we
are
in
the
situation
of
[IUTchIII],
Corollary
3.12,
and
elliptic
curve
E
F
has
that
the
good
non
V(F
)
that
does
not
divide
good
reduction
at
every
valuation
∈
V(F
)
def
2l.
In
the
notation
of
[IUTchI],
Definition
3.1,
let
us
write
d
mod
=
[F
mod
:
Q],
(1
≤)
e
mod
(≤
d
mod
)
for
the
maximal
ramification
index
of
F
mod
[i.e.,
of
valu-
def
def
∗
12
3
∗
12
3
∗
ations
∈
V
non
mod
]
over
Q,
d
mod
=
2
·
3
·
5
·
d
mod
,
e
mod
=
2
·
3
·
5
·
e
mod
(≤
d
mod
),
and
def
F
mod
⊆
F
tpd
=
F
mod
(
E
F
mod
[2]
)
⊆
F
for
the
“tripodal”
intermediate
field
obtained
from
F
mod
by
adjoining
the
fields
of
definition
of
the
2-torsion
points
of
any
model
of
E
F
×
F
F
over
F
mod
[cf.
Proposition
1.8,
(ii),
(iii)].
Moreover,
we
assume
that
the
(3·5)-torsion
points
of
E
F
are
defined
over
F
,
and
that
F
=
F
mod
(
√
−1,
E
F
mod
[2
·
3
·
5]
)
def
=
F
tpd
(
√
−1,
E
F
tpd
[3
·
5]
)
√
—
i.e.,
that
F
is
obtained
from
F
tpd
by
adjoining
−1,
together
with
the
fields
of
definition
of
the
(3
·
5)-torsion
points
of
a
model
E
F
tpd
of
the
elliptic
curve
E
F
×
F
F
over
F
tpd
determined
by
the
Legendre
form
of
the
Weierstrass
equation
[cf.,
e.g.,
the
statement
of
Corollary
2.2,
below;
Proposition
1.8,
(vi)].
[Thus,
it
follows
from
Proposition
1.8,
(iv),
that
E
F
∼
=
E
F
tpd
×
F
tpd
F
over
F
,
and
from
[IUTchI],
Definition
3.1,
(c),
that
l
=
5.]
If
F
mod
⊆
F
⊆
K
is
any
intermediate
extension
which
is
Galois
over
F
mod
,
then
we
shall
write
F
d
ADiv
∈
ADiv
R
(F
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
23
for
the
effective
arithmetic
divisor
determined
by
the
different
ideal
of
F
over
Q,
F
q
ADiv
∈
ADiv
R
(F
)
for
the
effective
arithmetic
divisor
determined
by
the
q-parameters
of
the
elliptic
def
curve
E
F
at
the
elements
of
V(F
)
bad
=
V
bad
mod
×
V
mod
V(F
)
(
=
∅)
[cf.
[GenEll],
Remark
3.3.1],
F
f
ADiv
∈
ADiv
R
(F
)
F
for
the
effective
arithmetic
divisor
whose
support
coincides
with
Supp(q
ADiv
),
but
all
of
whose
coefficients
are
equal
to
1
—
i.e.,
the
conductor
—
and
F
def
F
log(d
v
)
=
deg
V(F
)
v
(d
ADiv
)
∈
R
≥0
;
F
def
F
log(d
v
Q
)
=
deg
V(F
)
v
(d
ADiv
)
∈
R
≥0
Q
def
F
log(d
F
)
=
deg(d
ADiv
)
∈
R
≥0
def
F
log(q
v
)
=
deg
V(F
)
v
(q
ADiv
)
∈
R
≥0
;
def
def
F
log(q
v
Q
)
=
deg
V(F
)
v
(q
ADiv
)
∈
R
≥0
Q
F
log(q)
=
deg(q
ADiv
)
∈
R
≥0
def
F
F
log(f
v
)
=
deg
V(F
)
v
(f
ADiv
)
∈
R
≥0
;
def
F
def
F
log(f
v
Q
)
=
deg
V(F
)
v
(f
ADiv
)
∈
R
≥0
Q
F
log(f
F
)
=
deg(f
ADiv
)
∈
R
≥0
def
—
where
v
∈
V
mod
=
V(F
mod
),
v
Q
∈
V
Q
=
V(Q),
V(F
)
v
=
V(F
)
×
V
mod
{v},
def
V(F
)
v
Q
=
V(F
)
×
V
Q
{v
Q
}.
Here,
we
observe
that
the
various
“log(q
(−)
)’s”
are
independent
of
the
choice
of
F
,
and
that
the
quantity
“|log(q)|
∈
R
>0
”
defined
in
1
[IUTchIII],
Corollary
3.12,
is
equal
to
2l
·
log(q)
∈
R
[cf.
the
definition
of
“q
”
v
in
[IUTchI],
Example
3.2,
(iv)].
Then
one
may
take
the
constant
“C
Θ
∈
R”
of
[IUTchIII],
Corollary
3.12,
to
be
l+1
·
(1
+
12·d
l
mod
)
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
10
·
(e
∗
mod
·
l
+
η
prm
)
4·|log(q)|
1
12
−
6
·
(1
−
l
2
)
·
log(q)
−
1
and
hence,
by
applying
the
inequality
“C
Θ
≥
−1”
of
[IUTchIII],
Corollary
3.12,
conclude
that
1
6
·
log(q)
≤
(1
+
20·d
l
mod
)
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
20
·
(e
∗
mod
·
l
+
η
prm
)
≤
(1
+
20·d
l
mod
)
·
(log(d
F
)
+
log(f
F
))
+
20
·
(e
∗
mod
·
l
+
η
prm
)
—
where
η
prm
is
the
positive
real
number
of
Proposition
1.6.
Proof.
For
ease
of
reference,
we
divide
our
discussion
into
steps,
as
follows.
(i)
We
begin
by
recalling
the
following
elementary
identities
for
n
∈
N
≥1
:
(E1)
(E2)
1
n
1
n
n
m=1
n
m=1
m
=
1
2
(n
+
1);
m
2
=
1
6
(2n
+
1)(n
+
1).
24
SHINICHI
MOCHIZUKI
Also,
we
recall
the
following
elementary
facts:
(E3)
For
p
a
prime
number,
the
cardinality
|GL
2
(F
p
)|
of
GL
2
(F
p
)
is
given
by
|GL
2
(F
p
)|
=
p(p
+
1)(p
−
1)
2
.
(E4)
For
p
=
2,
3,
5,
the
expression
of
(E3)
may
be
computed
as
follows:
2
=
3·2
4
;
5(5+1)(5−1)
2
=
5·2
5
·3.
2(2+1)(2−1)
2
=
2·3;
3(3+1)(3−1)
√
(E5)
The
degree
of
the
extension
F
mod
(
−1
)/F
mod
is
≤
2.
(E6)
We
have:
0
≤
log(2)
≤
1,
1
≤
log(3)
≤
log(π)
≤
log(5)
≤
2.
(ii)
Next,
let
us
observe
that
the
inequality
log(d
F
tpd
)
+
log(f
F
tpd
)
≤
log(d
F
)
+
log(f
F
)
follows
immediately
from
Proposition
1.3,
(i),
and
the
various
definitions
involved.
On
the
other
hand,
the
inequality
log(d
F
)
+
log(f
F
)
≤
log(d
F
tpd
)
+
log(f
F
tpd
)
+
log(2
11
·
3
3
·
5
2
)
≤
log(d
F
tpd
)
+
log(f
F
tpd
)
+
21
follows
by
applying
Proposition
1.3,
(i),
at
the
primes
that
do
not
divide
2
·
3
·
5
[where
we
recall
that
the
extension
F/F
tpd
is
tamely
ramified
over
such
primes
—
cf.
Proposition
1.8,
(vi),
(vii)]
and
applying
Proposition
1.3,
(ii),
together
with
(E3),
(E4),
(E5),
(E6),
and
the
fact
that
we
have
a
natural
outer
inclusion
Gal(F/F
tpd
)
→
GL
2
(F
3
)
×
GL
2
(F
5
)
×
Z/2Z,
at
the
primes
that
divide
2
·
3
·
5.
In
a
similar
vein,
since
the
extension
K/F
is
tamely
ramified
at
the
primes
that
do
not
divide
l,
and
we
have
a
natural
outer
inclusion
Gal(K/F
)
→
GL
2
(F
l
),
the
inequality
log(d
K
)
≤
log(d
K
)
+
log(f
K
)
≤
log(d
F
)
+
log(f
F
)
+
2
·
log(l)
≤
log(d
F
tpd
)
+
log(f
F
tpd
)
+
2
·
log(l)
+
21
follows
immediately
from
Proposition
1.3,
(i),
(ii).
Finally,
for
later
reference,
we
observe
that
(1
+
4
l
)
·
log(d
K
)
≤
(1
+
4
l
)
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
2
·
log(l)
+
46
—
where
we
apply
the
estimates
log(l)
≤
12
and
1
+
4
l
≤
2,
both
of
which
may
be
l
regarded
as
consequences
of
the
fact
that
l
≥
5
[cf.
also
(E6)].
(iii)
If
F
tpd
⊆
F
⊆
K
is
any
intermediate
extension
which
is
Galois
over
F
mod
,
then
we
shall
write
V(F
)
dst
⊆
V(F
)
non
for
the
set
of
“distinguished”
nonarchimedean
valuations
v
∈
V(F
)
non
,
i.e.,
v
that
extend
to
a
valuation
∈
V(K)
non
that
ramifies
over
Q.
Now
observe
that
it
follows
immediately
from
Proposition
1.8,
(vi),
(vii),
together
with
our
assumption
on
V(F
)
good
V(F
)
non
,
that
(D0)
if
v
∈
V(F
tpd
)
non
does
not
divide
2
·
3
·
5
·
l
and,
moreover,
is
not
contained
F
tpd
in
Supp(q
ADiv
),
then
the
extension
K/F
tpd
is
unramified
over
v.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
25
Next,
let
us
recall
the
well-known
fact
that
the
determinant
of
the
Galois
rep-
resentation
determined
by
the
torsion
points
of
an
elliptic
curve
over
a
field
of
characteristic
zero
is
the
abelian
Galois
representation
determined
by
the
cyclo-
tomic
character.
In
particular,
it
follows
[cf.
the
various
definitions
involved]
that
K
contains
a
primitive
4
·
3
·
5
·
l-th
root
of
unity,
hence
is
ramified
over
Q
at
any
valuation
∈
V(K)
non
that
divides
2
·
3
·
5
·
l.
Thus,
one
verifies
immediately
[i.e.,
by
applying
(D0);
cf.
also
[IUTchI],
Definition
3.1,
(c)]
that
the
following
conditions
on
a
valuation
v
∈
V(F
)
non
are
equivalent:
(D1)
v
∈
V(F
)
dst
.
F
F
(D2)
The
valuation
v
either
divides
2
·
3
·
5
·
l
or
lies
in
Supp(q
ADiv
+
d
ADiv
).
(D3)
The
image
of
v
in
V(F
tpd
)
lies
in
V(F
tpd
)
dst
.
Let
us
write
non
V
dst
mod
⊆
V
mod
;
V
dst
⊆
V
non
Q
Q
for
the
respective
images
of
V(F
tpd
)
dst
in
V
mod
,
V
Q
and,
for
F
∗
∈
{F
mod
,
Q}
and
v
Q
∈
V
Q
,
def
∗
=
e
v
·
v
∈
ADiv
R
(F
∗
)
s
F
ADiv
v∈V(F
∗
)
dst
def
F
∗
∗
log(s
F
v
Q
)
=
deg
V(F
∗
)
v
(s
ADiv
)
∈
R
≥0
;
Q
s
≤
ADiv
=
def
w
Q
∈V(Q)
dst
ι
w
Q
log(p
w
Q
)
·
w
Q
≤
log(s
≤
v
Q
)
=
deg
V(Q)
v
(s
ADiv
)
∈
R
≥0
;
def
Q
def
∗
log(s
F
∗
)
=
deg(s
F
ADiv
)
∈
R
≥0
∈
ADiv
R
(Q)
log(s
≤
)
=
deg(s
≤
ADiv
)
∈
R
≥0
def
def
def
—
where
we
write
V(F
∗
)
v
Q
=
V(F
∗
)
×
V
Q
{v
Q
};
we
set
ι
w
Q
=
1
if
p
w
Q
≤
e
∗
mod
·
l,
def
ι
w
Q
=
0
if
p
w
Q
>
e
∗
mod
·
l.
Then
one
verifies
immediately
[again,
by
applying
(D0);
cf.
also
[IUTchI],
Definition
3.1,
(c)]
that
the
following
conditions
on
a
valuation
are
equivalent:
v
Q
∈
V
non
Q
(D4)
v
Q
∈
V
dst
Q
.
(D5)
The
valuation
v
Q
ramifies
in
K.
F
tpd
F
tpd
(D6)
Either
p
v
Q
|
2
·
3
·
5
·
l
or
v
Q
lies
in
the
image
of
Supp(q
ADiv
+
d
ADiv
).
F
F
(D7)
Either
p
v
Q
|
2
·
3
·
5
·
l
or
v
Q
lies
in
the
image
of
Supp(q
ADiv
+
d
ADiv
).
Here,
we
observe
in
passing
that,
for
v
∈
V(F
),
(R1)
log(e
v
)
≤
log(2
11
·
3
3
·
5
·
e
mod
·
l
4
)
if
v
divides
l,
F
(R2)
log(e
v
)
≤
log(2
11
·
3
3
·
5
·
e
mod
·
l)
if
v
divides
2
·
3
·
5
or
lies
in
Supp(q
ADiv
)
[hence
does
not
divide
l],
(R3)
log(e
v
)
≤
log(2
11
·
3
3
·
5
·
e
mod
)
if
v
does
not
divide
2
·
3
·
5
·
l
and,
F
moreover,
is
not
contained
in
Supp(q
ADiv
),
and
hence
that
26
SHINICHI
MOCHIZUKI
(R4)
if
e
v
≥
p
v
−
1
>
p
v
−
2,
then
p
v
≤
2
12
·
3
3
·
5
·
e
mod
·
l
=
e
∗
mod
·
l,
and
log(e
v
)
≤
−3
+
4
·
log(e
∗
mod
·
l)
—
cf.
(E3),
(E4),
(E5),
(E6);
(D0);
Proposition
1.8,
(v),
(vii);
[IUTchI],
Definition
3.1,
(c).
Next,
for
later
reference,
we
observe
that
the
inequality
F
mod
1
)
p
v
Q
·
log(s
v
Q
≤
1
p
v
Q
·
log(p
v
Q
)
holds
for
any
v
Q
∈
V
Q
;
in
particular,
when
p
v
Q
=
l
(≥
5),
it
holds
that
F
mod
1
)
p
v
Q
·
log(s
v
Q
≤
1
p
v
Q
·
log(p
v
Q
)
≤
1
2
—
cf.
(E6).
On
the
other
hand,
it
follows
immediately
from
Proposition
1.3,
(i),
mod
by
considering
the
various
possibilities
for
elements
∈
Supp(s
F
ADiv
),
that
F
F
mod
log(s
F
)
≤
2
·
(log(d
v
Q
tpd
)
+
log(f
v
Q
tpd
))
v
Q
—
and
hence
that
F
mod
1
)
p
v
Q
·
log(s
v
Q
≤
F
tpd
F
tpd
2
p
v
Q
·
(log(d
v
Q
)
+
log(f
v
Q
))
—
for
any
v
Q
∈
V
Q
such
that
p
v
Q
∈
{2,
3,
5,
l}.
In
a
similar
vein,
we
conclude
that
log(s
Q
)
≤
2
·
d
mod
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
log(2
·
3
·
5
·
l)
≤
2
·
d
mod
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
5
+
log(l)
and
hence
that
Q
4
l
·
log(s
)
≤
8·d
mod
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
6
l
—
cf.
(E6);
the
fact
that
l
≥
5.
Combining
this
last
inequality
with
the
inequality
of
the
final
display
of
Step
(ii)
yields
the
inequality
(1
+
4
l
)
·
log(d
K
)
+
4
l
·
log(s
Q
)
≤
(1
+
12·d
l
mod
)
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
2
·
log(l)
+
52
—
where
we
apply
the
estimate
d
mod
≥
1.
(iv)
In
order
to
estimate
the
constant
“C
Θ
”
of
[IUTchIII],
Corollary
3.12,
we
must,
according
to
the
various
definitions
given
in
the
statement
of
[IUTchIII],
Corollary
3.12,
compute
an
upper
bound
for
the
procession-normalized
mono-analytic
log-volume
of
the
holomorphic
hull
of
the
union
of
the
possible
images
of
a
Θ-pilot
object,
relative
to
the
relevant
Kummer
isomorphisms
[cf.
[IUTchIII],
Theorem
3.11,
(ii)],
in
the
multiradial
representation
of
[IUTchIII],
Theorem
3.11,
(i),
which
we
regard
as
subject
to
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)
described
in
[IUTchIII],
Theorem
3.11,
(i),
(ii).
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
27
Thus,
we
proceed
to
estimate
this
log-volume
at
each
v
Q
∈
V
Q
.
Once
one
fixes
v
Q
,
this
amounts
to
estimating
the
component
of
this
log-volume
in
±
“I
Q
(
S
j+1
;n,◦
D
v
Q
)”
[cf.
the
notation
of
[IUTchIII],
Theorem
3.11,
(i),
(a)],
for
each
j
∈
{1,
.
.
.
,
l
},
which
we
shall
also
regard
as
an
element
of
F
l
,
and
then
computing
the
average,
over
j
∈
{1,
.
.
.
,
l
},
of
these
estimates.
Here,
we
recall
[cf.
[IUTchI],
Proposition
6.9,
(i);
[IUTchIII],
Proposition
3.4,
(ii)]
that
S
±
j+1
=
{0,
1,
.
.
.
,
j}.
Also,
we
recall
±
from
[IUTchIII],
Proposition
3.2,
that
“I
Q
(
S
j+1
;n,◦
D
v
Q
)”
is,
by
definition,
a
tensor
product
of
j
+
1
copies,
indexed
by
the
elements
of
S
±
j+1
,
of
the
direct
sum
of
the
Q-
spans
of
the
log-shells
associated
to
each
of
the
elements
of
V(F
mod
)
v
Q
[cf.,
especially,
the
second
and
third
displays
of
[IUTchIII],
Proposition
3.2].
In
particular,
for
each
collection
{v
i
}
i∈S
±
j+1
of
[not
necessarily
distinct!]
elements
of
V(F
mod
)
v
Q
,
we
must
estimate
the
com-
ponent
of
the
log-volume
in
question
corresponding
to
the
tensor
product
of
the
Q-spans
of
the
log-shells
associated
to
this
collection
{v
i
}
i∈S
±
and
then
compute
j+1
the
weighted
average
[cf.
the
discussion
of
Remark
1.7.1],
over
possible
collections
{v
i
}
i∈S
±
,
of
these
estimates.
j+1
(v)
Let
v
Q
∈
V
dst
Q
.
Fix
j,
{v
i
}
i∈S
±
j+1
∼
as
in
Step
(iv).
Write
v
i
∈
V
→
V
mod
=
V(F
mod
)
for
the
element
corresponding
to
v
i
.
We
would
like
to
apply
Proposition
1.4,
(iii),
to
the
present
situation,
by
taking
·
“I”
to
be
S
±
j+1
;
∗
·
“I
⊆
I”
to
be
the
set
of
i
∈
I
such
that
e
v
i
>
p
v
Q
−
2;
·
“k
i
”
to
be
K
v
i
[so
“R
i
”
will
be
the
ring
of
integers
O
K
vi
of
K
v
i
];
·
“i
†
”
to
be
j
∈
S
±
j+1
;
·
“λ”
to
be
0
if
v
j
∈
V
good
;
2
·
“λ”
to
be
“ord(−)”
of
the
element
q
j
[cf.
the
definition
of
“q
”
in
[IUTchI],
Example
3.2,
(iv)]
if
v
j
∈
V
v
j
bad
v
.
Thus,
the
inclusion
“φ(p
λ
·
(R
I
)
∼
)
⊆
p
λ−d
I
−a
I
·
log
p
(R
I
×
)”
of
Proposition
1.4,
(iii),
implies
that
the
result
of
multiplying
“p
λ−|I|
·
2
−|I|
·
log
p
(R
I
×
)”
by
a
suitable
nonpositive
[cf.
the
inequalities
concerning
“d
I
+a
I
”
that
constitute
the
final
portion
of
Proposition
1.4,
(iii)]
integer
power
of
p
v
Q
contains
the
“union
of
possible
images
of
a
Θ-pilot
object”
discussed
in
Step
(iv).
That
is
to
say,
the
indeterminacies
(Ind1)
and
(Ind2)
are
taken
into
account
by
the
arbitrary
nature
of
the
automorphism
“φ”
[cf.
Proposition
1.2],
while
the
indeterminacy
(Ind3)
is
taken
into
account
by
the
fact
that
we
are
considering
upper
bounds
[cf.
the
discussion
of
Step
(x)
of
the
proof
of
[IUTchIII],
Corollary
3.12],
together
with
the
fact
that
the
above-mentioned
integer
power
of
p
v
Q
is
nonpositive,
which
implies
that
the
module
obtained
by
multiplying
by
this
power
of
p
v
Q
contains
“p
λ−|I|
·
2
−|I|
·
log
p
(R
I
×
)”.
Thus,
an
upper
bound
on
the
component
of
the
log-volume
of
the
holomorphic
hull
under
28
SHINICHI
MOCHIZUKI
consideration
may
be
obtained
by
computing
an
upper
bound
for
the
log-volume
of
the
right-hand
side
of
the
inclusion
“p
λ−d
I
−a
I
·log
p
(R
I
×
)
⊆
p
λ−d
I
−a
I
−b
I
·(R
I
)
∼
”
of
Proposition
1.4,
(iii).
Such
an
upper
bound
“
−
λ
+
d
I
+
1
·
log(p)
+
{3
+
log(e
i
)}”
i∈I
∗
is
given
in
the
second
displayed
inequality
of
Proposition
1.4,
(iii).
Here,
we
note
that
if
e
v
i
≤
p
v
Q
−
2
for
all
i
∈
I,
then
this
upper
bound
assumes
the
form
“
−
λ
+
d
I
+
1
·
log(p)”.
On
the
other
hand,
by
(R4),
if
e
v
i
>
p
v
Q
−
2
for
some
i
∈
I,
then
it
follows
that
p
v
Q
≤
e
∗
mod
·
l,
and
log(e
v
i
)
≤
−3
+
4
·
log(e
∗
mod
·
l),
so
the
upper
bound
in
question
may
be
taken
to
be
∗
“
−
λ
+
d
I
+
1
·
log(p)
+
4(j
+
1)
·
l
mod
”
def
∗
—
where
we
write
l
mod
=
log(e
∗
mod
·
l).
Also,
we
note
that,
unlike
the
other
terms
that
appear
in
these
upper
bounds,
“λ”
is
asymmetric
with
respect
to
the
choice
of
“i
†
∈
I”
in
S
±
j+1
.
Since
we
would
like
to
compute
weighted
averages
[cf.
the
discussion
of
Remark
1.7.1],
we
thus
observe
that,
after
symmetrizing
with
respect
to
the
choice
of
“i
†
∈
I”
in
S
±
j+1
,
this
upper
bound
may
be
written
in
the
form
“β
e
”
[cf.
the
notation
of
Proposition
1.7]
if,
in
the
situation
of
Proposition
1.7,
one
takes
·
“E”
to
be
V(F
mod
)
v
Q
;
·
“n”
to
be
j
+
1,
so
an
element
“
e
∈
E
n
”
corresponds
precisely
to
a
collection
{v
i
}
i∈S
±
;
j+1
·
“λ
e
”,
for
an
element
e
∈
E
corresponding
to
v
∈
V(F
mod
)
=
V
mod
,
to
be
[(F
mod
)
v
:
Q
v
Q
]
∈
R
>0
;
·
“β
e
”,
for
an
element
e
∈
E
corresponding
to
v
∈
V(F
mod
)
=
V
mod
,
to
be
2
j
∗
1
log(d
K
v
)
−
2l(j+1)
·
log(q
v
)
+
j+1
·
log(p
v
Q
)
+
4
·
ι
v
Q
·
l
mod
def
def
—
where
we
recall
that
ι
v
Q
=
1
if
p
v
Q
≤
e
∗
mod
·
l,
ι
v
Q
=
0
if
p
v
Q
>
e
∗
mod
·
l.
Here,
we
note
that
it
follows
immediately
from
the
first
equality
of
the
first
dis-
play
of
Proposition
1.7
that,
after
passing
to
weighted
averages,
the
operation
of
symmetrizing
with
respect
to
the
choice
of
“i
†
∈
I”
in
S
±
j+1
does
not
affect
the
com-
putation
of
the
upper
bound
under
consideration.
Thus,
by
applying
Proposition
1.7,
we
obtain
that
the
resulting
“weighted
average
upper
bound”
is
given
by
2
j
Q
∗
≤
(j
+
1)
·
log(d
K
v
Q
)
−
2l
·
log(q
v
Q
)
+
log(s
v
Q
)
+
4(j
+
1)
·
l
mod
·
log(s
v
Q
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
29
—
where
we
recall
the
notational
conventions
introduced
in
Step
(iii).
Thus,
it
remains
to
compute
the
average
over
j
∈
F
l
.
By
averaging
over
j
∈
{1,
.
.
.
,
l
=
l−1
2
}
and
applying
(E1),
(E2),
we
obtain
the
“procession-normalized
upper
bound”
(l
+3)
(2l
+1)(l
+1)
∗
≤
·
log(d
K
·
log(q
v
Q
)
+
log(s
Q
v
Q
)
−
v
Q
)
+
2(l
+
3)
·
l
mod
·
log(s
v
Q
)
2
12l
K
Q
∗
≤
l+1
=
l+5
4
·
log(d
v
Q
)
−
24
·
log(q
v
Q
)
+
log(s
v
Q
)
+
(l
+
5)
·
l
mod
·
log(s
v
Q
)
≤
l+1
4
·
Q
∗
≤
1
4
20
(1
+
4
l
)
·
log(d
K
v
Q
)
−
6
·
log(q
v
Q
)
+
l
·
log(s
v
Q
)
+
3
·
l
mod
·
log(s
v
Q
)
—
where,
in
the
passage
to
the
final
displayed
inequality,
we
apply
the
estimates
4(l+5)
1
1
20
l+1
≤
l
and
l+1
≤
3
,
both
of
which
may
be
regarded
as
consequences
of
the
fact
that
l
≥
5.
\
V
dst
(vi)
Next,
let
v
Q
∈
V
non
Q
Q
.
Fix
j,
{v
i
}
i∈S
±
as
in
Step
(iv).
Write
j+1
∼
v
i
∈
V
→
V
mod
=
V(F
mod
)
for
the
element
corresponding
to
v
i
.
We
would
like
to
apply
Proposition
1.4,
(iv),
to
the
present
situation,
by
taking
·
“I”
to
be
S
±
j+1
;
·
“k
i
”
to
be
K
v
i
[so
“R
i
”
will
be
the
ring
of
integers
O
K
vi
of
K
v
i
].
dst
Here,
we
note
that
our
assumption
that
v
Q
∈
V
non
Q
\V
Q
implies
that
the
hypotheses
of
Proposition
1.4,
(iv),
are
satisfied.
Thus,
the
inclusion
“φ((R
I
)
∼
)
⊆
(R
I
)
∼
”
of
Proposition
1.4,
(iv),
implies
that
the
tensor
product
of
log-shells
under
consider-
ation
contains
the
“union
of
possible
images
of
a
Θ-pilot
object”
discussed
in
Step
(iv).
That
is
to
say,
the
indeterminacies
(Ind1)
and
(Ind2)
are
taken
into
account
by
the
arbitrary
nature
of
the
automorphism
“φ”
[cf.
Proposition
1.2],
while
the
indeterminacy
(Ind3)
is
taken
into
account
by
the
fact
that
we
are
considering
upper
bounds
[cf.
the
discussion
of
Step
(x)
of
the
proof
of
[IUTchIII],
Corollary
3.12],
together
with
the
fact
that
the
“container
of
possible
images”
is
precisely
equal
to
the
tensor
product
of
log-shells
under
consideration.
Thus,
an
upper
bound
on
the
component
of
the
log-volume
under
consideration
may
be
obtained
by
computing
an
upper
bound
for
the
log-volume
of
the
right-hand
side
“(R
I
)
∼
”
of
the
above
inclusion.
Such
an
upper
bound
“0”
is
given
in
the
final
equality
of
Proposition
1.4,
(iv).
One
may
then
compute
a
“weighted
average
upper
bound”
and
then
a
“procession-normalized
upper
bound”,
as
was
done
in
Step
(v).
The
resulting
“procession-normalized
upper
bound”
is
clearly
equal
to
0.
(vii)
Next,
let
v
Q
∈
V
arc
Q
.
Fix
j,
{v
i
}
i∈S
±
∼
j+1
as
in
Step
(iv).
Write
v
i
∈
V
→
V
mod
=
V(F
mod
)
for
the
element
corresponding
to
v
i
.
We
would
like
to
apply
Proposition
1.5,
(iii),
(iv),
to
the
present
situation,
by
taking
·
“I”
to
be
S
±
j+1
[so
|I|
=
j
+
1];
·
“V
”
to
be
V(F
mod
)
v
Q
;
30
SHINICHI
MOCHIZUKI
∼
·
“C
v
”
to
be
K
v
,
where
we
write
v
∈
V
→
V
mod
for
the
element
determined
by
v
∈
V
.
Then
it
follows
from
Proposition
1.5,
(iii),
(iv),
that
π
j+1
·
B
I
serves
as
a
container
for
the
“union
of
possible
images
of
a
Θ-pilot
object”
discussed
in
Step
(iv).
That
is
to
say,
the
indeterminacies
(Ind1)
and
(Ind2)
are
taken
into
account
by
the
fact
that
B
I
⊆
M
I
is
preserved
by
arbitrary
automorphisms
of
the
type
discussed
in
Proposition
1.5,
(iii),
while
the
indeterminacy
(Ind3)
is
taken
into
account
by
the
fact
that
we
are
considering
upper
bounds
[cf.
the
discussion
of
Step
(x)
of
the
proof
of
[IUTchIII],
Corollary
3.12],
together
with
the
fact
that,
by
Proposition
1.5,
(iv),
together
with
our
choice
of
the
factor
π
j+1
,
this
“container
of
possible
images”
contains
the
elements
of
M
I
obtained
by
forming
the
tensor
prod-
uct
of
elements
of
the
log-shells
under
consideration.
Thus,
an
upper
bound
on
the
component
of
the
log-volume
under
consideration
may
be
obtained
by
computing
an
upper
bound
for
the
log-volume
of
this
container.
Such
an
upper
bound
(j
+
1)
·
log(π)
follows
immediately
from
the
fact
that
[in
order
to
ensure
compatibility
with
arith-
metic
degrees
of
arithmetic
line
bundles
—
cf.
[IUTchIII],
Proposition
3.9,
(iii)
—
one
is
obliged
to
adopt
normalizations
which
imply
that]
the
log-volume
of
B
I
is
equal
to
0.
One
may
then
compute
a
“weighted
average
upper
bound”
and
then
a
“procession-normalized
upper
bound”,
as
was
done
in
Step
(v).
The
resulting
“procession-normalized
upper
bound”
is
given
by
l+5
4
·
log(π)
≤
l+1
4
·
4
—
cf.
(E1),
(E6);
the
fact
that
l
≥
5.
(viii)
Now
we
return
to
the
discussion
of
Step
(iv).
In
order
to
compute
the
desired
upper
bound
for
“C
Θ
”,
it
suffices
to
sum
over
v
Q
∈
V
Q
the
various
local
“procession-normalized
upper
bounds”
obtained
in
Steps
(v),
(vi),
(vii)
for
v
Q
∈
V
Q
.
By
applying
the
inequality
of
the
final
display
of
Step
(iii),
we
thus
obtain
1
the
following
upper
bound
for
“C
Θ
·|log(q)|”,
i.e.,
the
product
of
“C
Θ
”
and
2l
·log(q):
l+1
4
·
(1
+
12·d
l
mod
)
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
2
·
log(l)
+
56
−
16
·
(1
−
12
l
2
)
·
log(q)
∗
≤
+
20
3
·
l
mod
·
log(s
)
1
−
2l
·
log(q)
1
12
1
—
where
we
apply
the
estimate
l+1
4
·
6
·
l
2
≥
2l
[cf.
the
fact
that
l
≥
1].
Now
let
us
recall
the
constant
“η
prm
”
of
Proposition
1.6.
By
applying
Propo-
sition
1.6,
we
compute:
e
∗
∗
mod
·l
l
mod
·
log(s
≤
)
≤
log(e
∗
mod
·
l)
·
1
≤
43
·
log(e
∗
mod
·
l)
·
log(e
∗
·l)
mod
p
≤
e
∗
·l
mod
=
∗
4
3
·
e
mod
·
l
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
31
—
where
the
sum
ranges
over
the
primes
p
≤
e
∗
mod
·
l
—
if
e
∗
mod
·
l
≥
η
prm
;
∗
l
mod
·
log(s
≤
)
≤
log(e
∗
mod
·
l)
·
1
≤
p
≤
e
∗
·l
mod
=
η
prm
4
3
·
log(η
prm
)
·
log(η
prm
)
4
3
·
η
prm
—
where
the
sum
ranges
over
the
primes
p
≤
e
∗
mod
·
l
—
if
e
∗
mod
·
l
<
η
prm
.
Thus,
we
conclude
that
∗
l
mod
·
log(s
≤
)
≤
43
·
(e
∗
mod
·
l
+
η
prm
)
[i.e.,
regardless
of
the
size
of
e
∗
mod
·
l].
Also,
let
us
observe
that
∗
1
4
3
·
3
·
(e
mod
·
l
+
η
prm
)
≥
∗
1
4
3
·
3
·
e
mod
·
l
≥
2
·
2
·
2
12
·
3
·
5
·
l
≥
2
·
log(l)
+
56
—
where
we
apply
the
estimates
e
mod
≥
1,
2
12
·
3
·
5
≥
56,
l
≥
5
≥
1,
l
≥
log(l)
[cf.
the
fact
that
l
≥
5].
Thus,
substituting
back
into
our
original
upper
bound
for
“C
Θ
·
|log(q)|”,
we
obtain
the
following
upper
bound
for
“C
Θ
”:
l+1
4·|log(q)|
·
(1
+
12·d
l
mod
)
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
10
·
(e
∗
mod
·
l
+
η
prm
)
)
·
log(q)
−
1
−
16
·
(1
−
12
2
l
·
43
=
7·4
—
where
we
apply
the
estimate
20+1
3
3
≤
10
—
i.e.,
as
asserted
in
the
statement
of
Theorem
1.10.
The
final
portion
of
Theorem
1.10
follows
immediately
from
[IUTchIII],
Corollary
3.12,
by
applying
the
inequality
of
the
first
display
of
Step
(ii),
together
with
the
estimates
−1
≤
2;
(1
−
12
l
2
)
−1
(1
−
12
·
(1
+
12·d
l
mod
)
≤
1
+
20·d
l
mod
l
2
)
[cf.
the
fact
that
l
≥
7,
d
mod
≥
1].
Remark
1.10.1.
One
of
the
main
original
motivations
for
the
development
of
the
theory
discussed
in
the
present
series
of
papers
was
to
create
a
framework,
or
geometry,
within
which
a
suitable
analogue
of
the
scheme-theoretic
Hodge-Arakelov
theory
of
[HASurI],
[HASurII]
could
be
realized
in
such
a
way
that
the
obstruc-
tions
to
diophantine
applications
that
arose
in
the
scheme-theoretic
formulation
of
[HASurI],
[HASurII]
[cf.
the
discussion
of
[HASurI],
§1.5.1;
[HASurII],
Remark
3.7]
could
be
avoided.
From
this
point
of
view,
it
is
of
interest
to
observe
that
the
com-
putation
of
the
“leading
term”
of
the
inequality
of
the
final
display
of
the
statement
of
Theorem
1.10
—
i.e.,
of
the
term
(l
+3)
(2l
+1)(l
+1)
·
log(d
K
·
log(q
v
Q
)
v
Q
)
−
2
12l
that
occurs
in
the
final
display
of
Step
(v)
of
the
proof
of
Theorem
1.10
—
via
the
identities
(E1),
(E2)
is
essentially
identical
to
the
computation
of
the
leading
term
that
occurs
in
the
proof
of
[HASurI],
Theorem
A
[cf.
the
discussion
following
the
statement
of
Theorem
A
in
[HASurI],
§1.1].
That
is
to
say,
in
some
sense,
32
SHINICHI
MOCHIZUKI
the
computations
performed
in
the
proof
of
Theorem
1.10
were
already
essentially
known
to
the
author
around
the
year
2000;
the
problem
then
was
to
construct
an
appropriate
framework,
or
geometry,
in
which
these
computations
could
be
performed!
This
sort
of
situation
may
be
compared
to
the
computations
underlying
the
Weil
Conjectures
priori
to
the
construction
of
a
“Weil
cohomology”
in
which
those
computations
could
be
performed,
or,
alternatively,
to
various
computations
of
invariants
in
topology
or
differential
geometry
that
were
motivated
by
computations
in
physics,
again
prior
to
the
construction
of
a
suitable
mathematical
framework
in
which
those
computations
could
be
performed.
Remark
1.10.2.
The
computation
performed
in
the
proof
of
Theorem
1.10
may
be
thought
of
as
the
computation
of
a
sort
of
derivative
in
the
F
l
-direction,
which,
relative
to
the
analogy
between
the
theory
of
the
present
series
of
papers
and
the
p-adic
Teichmüller
theory
of
[pOrd],
[pTeich],
corresponds
to
the
derivative
of
the
canonical
Frobenius
lifting
—
cf.
the
discussion
of
[IUTchIII],
Remark
3.12.4,
(iii).
In
this
context,
it
is
useful
to
recall
the
arithmetic
Kodaira-Spencer
morphism
that
occurs
in
scheme-theoretic
Hodge-Arakelov
theory
[cf.
[HASurII],
§3].
In
particular,
in
[HASurII],
Corollary
3.6,
it
is
shown
that,
when
suitably
formulated,
a
“certain
portion”
of
this
arithmetic
Kodaira-Spencer
morphism
coincides
with
the
usual
geometric
Kodaira-Spencer
morphism.
From
the
point
of
view
of
the
action
of
GL
2
(F
l
)
on
the
l-torsion
points
involved,
this
“certain
portion”
consists
of
the
unipotent
matrices
1
∗
0
1
of
GL
2
(F
l
).
By
contrast,
the
F
l
-symmetries
that
occur
in
the
present
series
of
papers
correspond
to
the
toral
matrices
∗
0
0
∗
of
GL
2
(F
l
)
—
cf.
the
discussion
of
[IUTchI],
Example
4.3,
(i).
As
we
shall
see
in
§2
below,
in
the
present
series
of
papers,
we
shall
ultimately
take
l
to
be
“large”.
When
l
is
“sufficiently
large”,
GL
2
(F
l
)
may
be
thought
of
as
a
“good
approximation”
for
GL
2
(Z)
or
GL
2
(R)
—
cf.
the
discussion
of
[IUTchI],
Remark
6.12.3,
(i),
(iii).
In
the
case
of
GL
2
(R),
“toral
subgroups”
may
be
thought
of
as
corresponding
to
the
isotropy
subgroups
[isomorphic
to
S
1
]
of
points
that
arise
from
the
action
of
GL
2
(R)
on
the
upper
half-plane,
i.e.,
subgroups
which
may
be
thought
of
as
a
sort
of
geometric,
group-theoretic
representation
of
tangent
vectors
at
a
point.
Remark
1.10.3.
The
“terms
involving
l”
that
occur
in
the
inequality
of
the
final
display
of
Theorem
1.10
may
be
thought
of
as
an
inevitable
consequence
of
the
fundamental
role
played
in
the
theory
of
the
present
series
of
papers
by
the
l-torsion
points
of
the
elliptic
curve
under
consideration.
Here,
we
note
that
it
is
of
crucial
importance
to
work
over
the
field
of
rationality
of
the
l-torsion
points
[i.e.,
“K”
as
opposed
to
“F
”]
not
only
when
considering
the
global
portions
of
the
various
ΘNF-
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
33
and
Θ
±ell
-Hodge
theaters
involved,
but
also
when
considering
the
local
portions
—
i.e.,
the
prime-strips
—
of
these
ΘNF-
and
Θ
±ell
-Hodge
theaters.
That
is
to
say,
these
local
portions
are
necessary,
for
instance,
in
order
to
glue
together
the
ΘNF-
and
Θ
±ell
-Hodge
theaters
that
appear
so
as
to
form
a
Θ
±ell
NF-Hodge
theater
[cf.
the
discussion
of
[IUTchI],
Remark
6.12.2].
In
particular,
to
allow,
within
these
local
portions,
any
sort
of
“Galois
indeterminacy”
with
respect
to
the
l-torsion
good
non
V
,
which,
at
first
glance,
might
points
—
even,
for
instance,
at
v
∈
V
appear
irrelevant
to
the
theory
of
Hodge-Arakelov-theoretic
evaluation
at
l-torsion
points
developed
in
[IUTchII]
—
would
have
the
effect
of
invalidating
the
various
delicate
manipulations
involving
l-torsion
points
discussed
in
[IUTchI],
§4,
§6
[cf.,
e.g.,
[IUTchI],
Propositions
4.7,
6.5].
Remark
1.10.4.
The
various
fluctuations
in
log-volume
—
i.e.,
whose
computa-
tion
is
the
subject
of
Theorem
1.10!
—
that
arise
from
the
multiradial
representation
of
[IUTchIII],
Theorem
3.11,
(i),
may
be
thought
of
as
a
sort
of
“inter-universal
analytic
torsion”.
Indeed,
in
general,
“analytic
torsion”
may
be
understood
as
a
sort
of
measure
—
in
“metrized”
[e.g.,
log-volume!]
terms
—
of
the
degree
of
deviation
of
the
“holomorphic
functions”
[such
as
sections
of
a
line
bundle]
on
a
variety
—
i.e.,
which
depend,
in
an
essential
way,
on
the
holomorphic
moduli
of
the
variety!
—
from
the
“real
analytic
functions”
—
i.e.,
which
are
invariant
with
respect
to
deformations
of
the
holomorphic
moduli
of
the
variety.
For
instance:
(a)
In
“classical”
Arakelov
theory,
analytic
torsion
typically
arises
as
[the
log-
arithm
of]
a
sort
of
normalized
determinant
of
the
Laplacian
acting
on
some
space
of
real
analytic
[or
L
2
-]
sections
of
a
line
bundle
on
a
complex
variety
equipped
with
a
real
analytic
Kähler
metric
[cf.,
e.g.,
[Arak],
Chapters
V,
VI].
Here,
we
recall
that
in
this
sort
of
situation,
the
space
of
holomorphic
sections
of
the
line
bundle
is
given
by
the
kernel
of
the
Laplacian;
the
definition
of
the
Laplacian
depends,
in
an
essential
way,
on
the
Kähler
metric,
hence,
in
particular,
on
the
holomorphic
moduli
of
the
variety
under
consideration
[cf.,
e.g.,
the
case
of
the
Poincaré
metric
on
a
hyperbolic
Riemann
surface!].
(b)
In
the
scheme-theoretic
Hodge-Arakelov
theory
discussed
in
[HASurI],
[HA-
SurII],
the
main
theorem
consists
of
a
sort
of
comparison
isomorphism
[cf.
[HASurI],
Theorem
A]
between
a
certain
subspace
of
the
space
of
global
sections
of
the
pull-
back
of
an
ample
line
bundle
on
an
elliptic
curve
to
the
universal
vectorial
extension
of
the
elliptic
curve
and
the
space
of
set-theoretic
functions
on
the
torsion
points
of
the
elliptic
curve.
That
is
to
say,
the
former
space
of
sections
contains,
in
a
natu-
ral
way,
the
space
of
holomorphic
sections
of
the
ample
line
bundle
on
the
elliptic
curve,
while
the
latter
space
of
functions
may
be
thought
of
as
a
sort
of
“discrete
approximation”
of
the
space
of
real
analytic
functions
on
the
elliptic
curve
[cf.
the
discussion
of
[HASurI],
§1.3.2,
§1.3.4].
In
this
context,
the
“Gaussian
poles”
[cf.
the
discussion
of
[HASurI],
§1.1]
arise
as
a
measure
of
the
discrepancy
of
integral
structures
between
these
two
spaces
in
a
neighborhood
of
the
divisor
at
infinity
of
34
SHINICHI
MOCHIZUKI
the
moduli
stack
of
elliptic
curves,
hence
may
be
thought
of
as
a
sort
of
“analytic
torsion
at
the
divisor
at
infinity”
[cf.
the
discussion
of
[HASurI],
§1.2].
(c)
In
the
case
of
the
multiradial
representation
of
[IUTchIII],
Theorem
3.11,
(i),
the
fluctuations
of
log-volume
computed
in
Theorem
1.10
arise
precisely
as
a
result
of
the
execution
of
a
comparison
of
an
“alien”
arithmetic
holomorphic
structure
to
this
multiradial
representation,
which
is
compatible
with
the
per-
mutation
symmetries
of
the
étale-picture,
i.e.,
which
is
“invariant
with
respect
to
deformations
of
the
arithmetic
holomorphic
moduli
of
the
number
field
under
con-
sideration”
in
the
sense
that
it
makes
sense
simultaneously
with
respect
to
distinct
arithmetic
holomorphic
structures
[cf.
[IUTchIII],
Remark
3.11.1;
[IUTchIII],
Re-
mark
3.12.3,
(ii)].
Here,
it
is
of
interest
to
observe
that
the
object
of
this
comparison
consists
of
the
values
of
the
theta
function,
i.e.,
in
essence,
a
“holomorphic
section
of
an
ample
line
bundle”.
In
particular,
the
resulting
fluctuations
of
log-volume
may
be
thought
as
a
sort
of
“analytic
torsion”.
By
analogy
to
the
terminology
“Gaussian
poles”
discussed
in
(b)
above,
it
is
natural
to
think
of
the
terms
involv-
ing
the
different
d
K
(−)
that
appear
in
the
computation
underlying
Theorem
1.10
[cf.,
e.g.,
the
final
display
of
Step
(v)
of
the
proof
of
Theorem
1.10]
as
“differential
poles”
[cf.
the
discussion
of
Remarks
1.10.1,
1.10.2].
Finally,
in
the
context
of
the
normalized
determinants
that
appear
in
(a),
it
is
interesting
to
note
the
role
played
by
the
prime
number
theorem
—
i.e.,
in
essence,
the
Riemann
zeta
func-
tion
[cf.
Proposition
1.6
and
its
proof]
—
in
the
computation
of
“inter-universal
analytic
torsion”
given
in
the
proof
of
Theorem
1.10.
Remark
1.10.5.
The
above
remarks
focused
on
the
conceptual
aspects
of
the
theory
surrounding
Theorem
1.10.
Before
proceeding,
however,
we
pause
to
discuss
briefly
certain
aspects
of
Theorem
1.10
that
are
of
interest
from
a
computational
point
of
view,
i.e.,
in
the
spirit
of
conventional
analytic
number
theory.
(i)
First,
we
begin
by
observing
that,
unlike
the
inequalities
that
appear
in
the
various
results
[cf.
Corollaries
2.2,
(ii);
2.3]
obtained
in
§2
below,
the
inequalities
obtained
in
Theorem
1.10
involve
only
essentially
explicit
constants
and,
more-
over,
do
not
require
one
to
exclude
some
non-explicit
finite
set
of
“isomorphism
classes
of
exceptional
elliptic
curves”.
From
this
point
of
view,
the
inequalities
obtained
in
Theorem
1.10
are
suited
to
application
to
computations
concerning
various
explicit
diophantine
equations,
such
as,
for
instance,
the
equations
that
appear
in
“Fermat’s
Last
Theo-
rem”.
Such
explicit
computations
in
the
case
of
specific
diophantine
equations
are,
how-
ever,
beyond
the
scope
of
the
present
paper.
(ii)
One
topic
of
interest
in
the
context
of
computational
aspects
of
Theorem
1.10
is
the
asymptotic
behavior
of
the
bound
that
appears
in,
say,
the
first
inequality
of
the
final
display
of
Theorem
1.10.
Let
us
assume,
for
simplicity,
def
that
F
tpd
=
Q
[so
d
mod
=
1].
Also,
to
simplify
the
notation,
let
us
write
δ
=
log(d
F
tpd
)
+
log(f
F
tpd
)
=
log(f
F
tpd
).
Then
the
bound
under
consideration
assumes
the
form
δ
+
∗
·
δl
+
∗
·
l
+
∗
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
35
—
where,
in
the
present
discussion,
the
“∗’s”
are
to
be
understood
as
denoting
fixed
positive
real
numbers.
Thus,
the
leading
term
[cf.
the
discussion
of
Remark
1.10.1]
is
equal
to
δ.
The
remaining
terms
give
rise
to
the
“
terms”
[and
bounded
discrep-
ancy]
of
the
inequalities
of
Corollaries
2.2,
(ii);
2.3,
obtained
in
§2
below.
Thus,
if
one
ignores
“bounded
discrepancies”,
it
is
of
interest
to
consider
the
behavior
of
the
“
terms”
∗
·
δl
+
∗
·
l
as
one
allows
the
initial
Θ-data
under
consideration
to
vary
[i.e.,
subject
to
the
condition
“F
tpd
=
Q”].
In
this
context,
one
fundamental
observation
is
the
fol-
lowing:
although
l
is
subject
to
various
other
conditions,
no
matter
how
“skillfully”
one
chooses
l,
the
resulting
“
terms”
are
always
≥
∗
·
δ
1/2
—
an
estimate
that
may
be
obtained
by
thinking
of
l
as
≈
δ
α
,
for
some
real
number
α,
and
comparing
δ
α
and
δ
1−α
.
This
estimate
is
of
particular
interest
in
the
context
of
various
explicit
examples
constructed
by
Masser
and
others
[cf.
[Mss];
the
discussion
of
[vFr],
§2]
in
which
explicit
“abc
sums”
are
constructed
for
which
the
quantity
on
the
left-hand
side
of
the
inequality
of
Theorem
1.10
under
consideration
exceeds
the
order
of
δ
+
1/2
δ
∗
·
log(δ)
—
cf.
[vFr],
Equation
(6).
In
particular,
the
asymptotic
estimates
given
by
Theorem
1.10
are
consistent
with
the
known
asymptotic
behavior
of
these
explicit
abc
sums.
Indeed,
the
exponent
“
12
”
that
appears
in
the
fundamental
observation
discussed
above
coincides
precisely
with
the
“expectation”
expressed
by
van
Frankenhuijsen
in
the
final
portion
of
the
discussion
of
[vFr],
§2!
In
the
present
paper,
although
we
are
unable
to
in
fact
achieve
bounds
on
the
“
terms”
of
the
order
∗
·
δ
1/2
,
we
do
succeed
in
obtaining
bounds
on
the
“
terms”
of
the
order
∗
·
δ
1/2
·
log(δ)
—
albeit
under
the
assumption
that
the
abc
sums
under
consideration
are
com-
pactly
bounded
away
from
infinity
at
the
prime
2,
as
well
as
at
the
archimedean
prime
[cf.
Corollary
2.2,
(ii);
Remark
2.2.1
below
for
more
details].
(iii)
In
the
context
of
the
discussion
of
(ii),
it
is
of
interest
to
observe
that
the
“∗
·
l”
portion
of
the
“
terms”
that
appear
arises
from
the
estimates
given
in
Step
(viii)
of
the
proof
of
Theorem
1.10
for
the
quantity
“log(s
≤
)”.
From
the
point
of
view
of
the
discussion
of
[vFr],
§3,
this
quantity
corresponds
essentially
to
a
“certain
portion”
of
the
quantity
“ω(abc)”
associated
to
an
abc
sum.
That
is
to
say,
whereas
“ω(abc)”
denotes
the
total
number
of
prime
factors
that
occur
in
the
product
abc,
the
quantity
“log(s
≤
)”
corresponds,
roughly
speaking,
to
the
number
of
these
prime
factors
that
are
≤
e
∗
mod
·
l.
The
appearance
[i.e.,
in
the
proof
of
Theorem
1.10]
of
such
a
term
which
is
closely
related
to
“ω(abc)”
is
of
interest
from
the
point
of
view
of
the
discussion
of
[vFr],
§3,
partly
since
it
is
[not
precisely
identical
to,
but
nonetheless]
reminiscent
of
the
various
refinements
of
the
ABC
Conjecture
proposed
by
Baker
[i.e.,
which
are
the
main
topic
of
the
discussion
of
36
SHINICHI
MOCHIZUKI
[vFr],
§3].
The
appearance
[i.e.,
in
the
proof
of
Theorem
1.10]
of
such
a
term
which
is
closely
related
to
“ω(abc)”
is
also
of
interest
from
the
point
of
view
of
the
explicit
δ
1/2
abc
sums
discussed
in
(ii)
that
give
rise
to
asymptotic
behavior
≥
∗
·
log(δ)
.
That
is
to
say,
according
to
the
discussion
of
[vFr],
§3,
Remark
1,
this
sort
of
abc
sum
tends
to
give
rise
to
a
relatively
large
value
for
ω(abc)
—
i.e.,
a
state
of
affairs
that
is
con-
sistent
with
the
crucial
role
played
by
the
“
term”
related
to
ω(abc)
in
the
computation
of
the
lower
bound
“≥
∗
·
δ
1/2
”
that
appears
in
the
fundamental
observation
of
(ii).
By
contrast,
the
abc
sums
of
the
form
“2
n
=
p
+
qr”
[where
p,
q,
and
r
are
prime
numbers]
considered
in
[vFr],
§3,
Remark
1,
give
rise
to
a
relatively
small
value
for
ω(abc)
[indeed,
ω(abc)
≤
4]
—
i.e.,
a
situation
that
suggests
relatively
small/essentially
negligible
“
terms”
in
the
bound
of
Theorem
1.10
under
consideration.
Such
essentially
negligible
“
terms”
are,
however,
consistent
with
the
fact
[cf.
[vFr],
§3,
Remark
1]
that,
for
such
abc
sums,
the
left-hand
side
of
the
inequality
of
Theorem
1.10
under
consideration
is
roughly
≈
12
·
the
leading
term
of
the
bound
on
the
right-hand
side,
hence,
in
particular,
is
amply
bounded
by
the
leading
term
on
the
right-hand
side,
without
any
“help”
from
the
“
terms”.
Remark
1.10.6.
(i)
In
the
context
of
the
discussion
of
Remark
1.10.5,
it
is
important
to
remem-
ber
that
the
bound
on
“
16
·
log(q)”
given
in
Theorem
1.10
only
concerns
the
q-
parameters
at
the
nonarchimedean
valuations
contained
in
V
bad
mod
,
all
of
which
are
necessarily
of
odd
residue
characteristic
—
cf.
[IUTchI],
Definition
3.1,
(b).
This
observation
is
of
relevance
to
the
examples
of
abc
sums
constructed
in
[Mss]
[cf.
the
discussion
of
Remark
1.10.5,
(ii)],
since
it
does
not
appear,
at
first
glance,
that
there
is
any
way
to
effectively
control
the
contributions
at
the
prime
2
in
these
examples,
that
is
to
say,
in
the
notation
of
the
Proposition
of
[Mss],
to
control
the
power
of
2
that
divides
the
integer
“c”
of
the
Proposition
of
[Mss],
or,
alternatively,
in
the
notation
of
the
proof
of
this
Proposition
on
[Mss],
p.
22,
to
control
the
power
of
2
that
divides
the
difference
“x
i
−
x
i−1
”.
On
the
other
hand,
it
was
pointed
out
to
the
author
by
A.
Venkatesh
that
in
fact
it
is
not
difficult
to
modify
the
construction
of
these
examples
of
abc
sums
given
in
[Mss]
so
as
to
obtain
similar
asymptotic
estimates
to
those
obtained
in
[Mss]
[cf.
the
discussion
of
Remark
1.10.5,
(ii)],
even
without
taking
into
account
the
contributions
at
the
prime
2.
(ii)
In
the
context
of
the
discussion
of
(i),
it
is
of
interest
to
recall
why
nonar-
chimedean
primes
of
even
residue
characteristic
where
the
elliptic
curve
under
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
37
consideration
has
bad
multiplicative
reduction
are
excluded
from
V
bad
mod
in
the
the-
ory
of
the
present
series
of
papers.
In
a
word,
the
reason
that
the
theory
encounters
difficulties
at
primes
over
2
is
that
it
depends,
in
a
quite
essential
way,
on
the
theory
of
the
étale
theta
function
developed
in
[EtTh],
which
fails
at
primes
over
2
[cf.
the
assumption
that
“p
is
odd”
in
[EtTh],
Theorem
1.10,
(iii);
[EtTh],
Definition
2.5;
[EtTh],
Corollary
2.18].
From
the
point
of
view
of
the
theory
of
[IUTchI],
[IUTchII],
and
[IUTchIII]
[cf.,
especially,
the
theory
of
[IUTchII],
§1,
§2:
[IUTchII],
Corollary
1.12;
[IUTchII],
Corollary
2.4,
(ii),
(iii);
[IUTchII],
Corollary
2.6],
one
of
the
key
consequences
of
the
theory
of
[EtTh]
is
the
simultaneous
multiradiality
of
the
algorithms
that
give
rise
to
(1)
constant
multiple
rigidity
and
(2)
cyclotomic
rigidity.
At
a
more
concrete
level,
(1)
is
obtained
by
evaluating
the
usual
series
for
the
theta
function
[cf.
[EtTh],
Proposition
1.4]
at
the
2-torsion
point
in
the
“irreducible
component
labeled
zero”.
One
computes
easily
that
the
resulting
“special
value”
is
a
unit
for
odd
p,
but
is
equal
to
a
[nonzero]
non-unit
when
p
=
2.
In
particular,
since
(1)
is
established
by
dividing
the
series
of
[EtTh],
Proposition
1.4
[i.e.,
the
usual
series
for
the
theta
function],
by
this
special
value,
it
follows
that
(a)
the
“integral
structure”
on
the
theta
function
determined
by
this
special
value
coincides
with
(b)
the
“integral
structure”
on
the
theta
function
determined
by
the
natural
integral
structure
on
the
pole
at
the
origin
for
odd
p
[cf.
[EtTh],
Theorem
1.10,
(iii)],
but
not
when
p
=
2.
That
is
to
say,
when
p
=
2,
a
nontrivial
denominator
arises.
Here,
we
recall
that
it
is
crucial
to
evaluate
at
2-torsion
points,
i.e.,
as
opposed
to,
say,
more
general
points
in
the
irreducible
component
labeled
zero
for
reasons
discussed
in
[IUTchII],
Remark
2.5.1,
(ii)
[cf.
also
the
discussion
of
[IUTchII],
Remark
1.12.2,
(i),
(ii),
(iii),
(iv)].
This
nontrivial
denominator
is
fundamentally
incompatible
with
the
multiradiality
of
the
algorithms
of
(1),
(2)
in
that
it
is
incompatible
with
the
fundamental
splitting,
or
“decoupling”,
into
“purely
radial”
[i.e.,
roughly
speaking,
“value
group”]
and
“purely
coric”
[i.e.,
roughly
speaking,
“unit”]
components
discussed
in
[IUTchII],
Remarks
1.11.4,
(i);
1.12.2,
(vi)
[cf.
also
the
discussion
of
[IUTchII],
Remark
1.11.5].
That
is
to
say,
on
the
one
hand,
the
multiradiality
of
(1)
may
only
be
established
if
the
possible
values
at
the
evaluation
points
in
the
irreducible
component
labeled
zero
are
known,
a
priori,
to
be
units,
i.e.,
if
one
works
relative
to
the
integral
structure
(a)
—
cf.
the
discussion
of
[IUTchII],
Remark
1.12.2,
(i),
(ii),
(iii),
(iv).
On
the
other
hand,
if
one
tries
to
work
38
SHINICHI
MOCHIZUKI
simultaneously
with
the
integral
structure
(b),
hence
with
the
non-
trivial
denominator
discussed
above,
then
the
multiradiality
of
(2)
is
violated.
Here,
we
recall
that
the
integral
structure
(b),
which
is
referred
to
as
the
“canonical
integral
structure”
in
[EtTh],
Proposition
1.4,
(iii);
[EtTh],
Theorem
1.10,
(iii),
is
in
some
sense
the
“integral
structure
of
common
sense”.
(iii)
It
is
not
entirely
clear
to
the
author
at
the
time
of
writing
to
what
extent
the
integral
structure
(b)
is
necessary
in
order
to
carry
out
the
theory
developed
in
the
present
series
of
papers.
Indeed,
[EtTh],
as
well
as
the
present
series
of
papers,
was
written
in
a
way
that
[unlike
the
discussion
of
(ii)!]
“takes
for
granted”
the
fact
that
the
two
integral
structures
(a),
(b)
discussed
above
coincide
for
odd
p,
i.e.,
in
a
way
which
identifies
these
two
integral
structures
and
hence
does
not
specify,
at
various
key
points
in
the
discussion,
whether
one
is
in
fact
working
with
integral
structure
(a)
or
with
integral
structure
(b).
On
the
other
hand,
if
it
is
indeed
the
case
that
not
only
the
integral
structure
(a),
but
also
the
integral
struc-
ture
(b)
plays
an
essential
role
in
the
present
series
of
papers,
then
it
follows
[cf.
the
discussion
of
(ii)!]
that
the
theory
of
the
present
series
of
papers
is
funda-
mentally
incompatible
with
the
inclusion
in
V
bad
mod
of
nonarchimedean
primes
of
even
residue
characteristic
where
the
elliptic
curve
under
consideration
has
bad
multiplicative
reduction.
(iv)
In
the
context
of
the
discussion
of
(ii),
(iii),
it
is
perhaps
useful
to
recall
that
the
classical
theory
of
theta
functions
also
tends
to
[depending
on
your
point
of
view!]
“break
down”
or
“assume
a
completely
different
form”
at
the
prime
2.
For
instance,
this
phenomenon
can
be
seen
throughout
Mumford’s
theory
of
algebraic
theta
functions,
which
may
be
thought
of
as
a
sort
of
predecessor
to
the
scheme-
theoretic
Hodge-Arakelov
theory
of
[HASurI],
[HASurII],
which,
in
turn,
may
be
thought
of
as
a
sort
of
predecessor
to
the
theory
of
the
present
series
of
papers.
In
a
similar
vein,
it
is
of
interest
to
recall
that
the
prime
2
is
also
excluded
in
the
p-adic
Teichmüller
theory
of
[pOrd],
[pTeich].
This
is
done
in
order
to
avoid
the
complications
that
occur
in
the
theory
of
the
Lie
algebra
sl
2
over
fields
of
characteristic
2.
Remark
1.10.7.
(i)
Since
e
∗
mod
≤
d
∗
mod
,
one
may
replace
“e
∗
mod
”
by
“d
∗
mod
”
in
the
final
two
displays
of
the
statement
of
Theorem
1.10.
(ii)
By
contrast,
at
least
if
one
adheres
to
the
framework
of
the
theory
of
the
present
series
of
papers,
it
is
not
possible
to
replace
“d
mod
”
by
“e
mod
”
in
the
final
two
displays
of
the
statement
of
Theorem
1.10.
The
fundamental
reason
for
this
is
that,
in
the
construction
of
the
multiradial
representation
of
[IUTchIII],
Theorem
3.11,
(i),
it
is
necessary
to
consider
tensor
products
of
copies,
labeled
by
j
∈
F
l
,
of
F
mod
over
Q
[cf.
[IUTchIII],
Proposition
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
39
3.3!].
That
is
to
say,
it
is
fundamentally
impossible
[i.e.,
relative
to
the
framework
of
the
theory
of
the
present
series
of
papers]
to
identify
the
F
mod
-linear
structures
for
distinct
labels
j,
since
the
various
tensor
packets
that
appear
in
the
multiradial
representation
must
be
constructed
in
such
a
way
as
to
depend
only
on
the
additive
structure
[i.e.,
not
the
module
structure
over
some
sort
of
ring
such
as
F
mod
!]
of
the
[mono-analytic!]
log-shells
involved.
Working
with
tensor
powers
of
copies
of
F
mod
over
Q
means
that
there
is
no
way
to
avoid,
when
one
localizes
at
a
prime
number
p,
working
with
tensor
products
between
localizations
of
F
mod
at
distinct
primes
of
F
mod
that
divide
p.
Moreover,
whenever
even
one
of
these
primes
of
F
mod
is
lies
under
a
prime
of
K
that
ramifies
over
Q
[cf.
condition
(D5)
of
Step
(iii)
of
the
proof
of
Theorem
1.10],
the
computation
of
Step
(v)
of
the
proof
of
Theorem
1.10
necessarily
gives
rise
to
a
“log(p)”
term
—
i.e.,
that
appears
in
“log(s
Q
)”
—
that
arises
from
“rounding
up”
non-integral
powers
of
p
[i.e.,
as
in
the
inclusions
of
Proposition
1.4,
(iii)],
since
only
integral
powers
of
p
make
sense
in
the
multiradial
representation.
That
is
to
say,
whereas
integral
powers
of
p
only
require
the
use
of
the
additive
structure
of
the
[mono-analytic!]
log-shells
involved,
non-integral
powers
only
make
sense
if
one
is
equipped
with
the
module
structure
over
some
sort
of
ring
such
as
F
mod
!
40
SHINICHI
MOCHIZUKI
Section
2:
Diophantine
Inequalities
In
the
present
§2,
we
combine
Theorem
1.10
with
the
theory
of
[GenEll]
to
give
a
proof
of
the
ABC
Conjecture,
or,
equivalently,
Vojta’s
Conjecture
for
hyperbolic
curves
[cf.
Corollary
2.3
below].
We
begin
by
reviewing
some
well-known
estimates.
Proposition
2.1.
(Well-known
Estimates)
(i)
(Linearization
of
Logarithms)
We
have
log(x)
≤
x
for
all
(R
)
x
≥
1.
(ii)
(The
Prime
Number
Theorem)
There
exists
a
real
number
ξ
prm
≥
5
such
that
def
2
·
x
≤
θ(x)
=
log(p)
≤
43
·
x
3
p≤x
—
where
the
sum
ranges
over
the
prime
numbers
p
such
that
p
≤
x
—
for
all
(R
)
x
≥
ξ
prm
.
In
particular,
if
A
is
a
finite
set
of
prime
numbers,
and
we
write
def
θ
A
=
log(p)
p∈A
[where
we
take
the
sum
to
be
0
if
A
=
∅],
then
there
exists
a
prime
number
p
∈
A
such
that
p
≤
2(θ
A
+
ξ
prm
).
Proof.
Assertion
(i)
is
well-known
and
entirely
elementary.
Assertion
(ii)
is
a
well-
known
consequence
of
the
Prime
Number
Theorem
[cf.,
e.g.,
[Edw],
p.
76;
[GenEll],
Lemma
4.1;
[GenEll],
Remark
4.1.1].
Let
Q
be
an
algebraic
closure
of
Q.
In
the
following
discussion,
we
shall
apply
the
notation
and
terminology
of
[GenEll].
Let
X
be
a
smooth,
proper,
geometrically
def
connected
curve
over
a
number
field;
D
⊆
X
a
reduced
divisor;
U
X
=
X\D;
d
a
positive
integer.
Write
ω
X
for
the
canonical
sheaf
on
X.
Suppose
that
U
X
is
a
hyperbolic
curve,
i.e.,
that
the
degree
of
the
line
bundle
ω
X
(D)
is
positive.
Then
we
recall
the
following
notation:
·
U
X
(Q)
≤d
⊆
U
X
(Q)
denotes
the
subset
of
Q-rational
points
defined
over
a
finite
extension
field
of
Q
of
degree
≤
d
[cf.
[GenEll],
Example
1.3,
(i)].
·
log-diff
X
denotes
the
[normalized]
log-different
function
on
U
X
(Q)
[cf.
[GenEll],
Definition
1.5,
(iii)].
·
log-cond
D
denotes
the
[normalized]
log-conductor
function
on
U
X
(Q)
[cf.
[GenEll],
Definition
1.5,
(iv)].
·
ht
ω
X
(D)
denotes
the
[normalized]
height
function
on
U
X
(Q)
associated
to
ω
X
(D),
which
is
well-defined
up
to
a
“bounded
discrepancy”
[cf.
[GenEll],
Proposition
1.4,
(iii)].
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
41
In
order
to
apply
the
theory
of
the
present
series
of
papers,
it
is
neceesary
to
construct
suitable
initial
Θ-data,
as
follows.
Corollary
2.2.
(Construction
of
Suitable
Initial
Θ-Data)
Suppose
that
X
=
P
1
Q
is
the
projective
line
over
Q,
and
that
D
⊆
X
is
the
divisor
consisting
of
the
three
points
“0”,
“1”,
and
“∞”.
We
shall
regard
X
as
the
“λ-line”
—
i.e.,
we
shall
regard
the
standard
coordinate
on
X
=
P
1
Q
as
the
“λ”
in
the
Legendre
form
“y
2
=
x(x−1)(x−λ)”
of
the
Weierstrass
equation
defining
an
elliptic
curve
—
and
hence
as
being
equipped
with
a
natural
classifying
morphism
U
X
→
(M
ell
)
Q
[cf.
the
discussion
preceding
Proposition
1.8].
Let
K
V
⊆
U
X
(Q)
be
a
compactly
bounded
subset
[i.e.,
regarded
as
a
subset
of
X(Q)
—
cf.
Re-
mark
2.3.1,
(vi),
below;
[GenEll],
Example
1.3,
(ii)]
whose
support
contains
the
nonarchimedean
prime
“2”.
Suppose
further
that
K
V
satisfies
the
following
condi-
tion:
(∗
j-inv
)
If
v
∈
V(Q)
denotes
the
nonarchimedean
prime
“2”,
then
the
image
of
the
subset
K
v
⊆
U
X
(Q
v
)
associated
to
K
V
[cf.
the
notational
conventions
of
[GenEll],
Example
1.3,
(ii)]
via
the
j-invariant
U
X
→
(M
ell
)
Q
→
A
1
Q
is
a
bounded
subset
of
A
1
Q
(Q
v
)
=
Q
v
,
i.e.,
is
contained
in
a
subset
of
the
form
2
N
j-inv
·
O
Q
⊆
Q
v
,
where
N
j-inv
∈
Z,
and
O
Q
⊆
Q
v
denotes
the
v
v
ring
of
integers.
Then:
2
(i)
Write
“log(q
∀
(−)
)”
(respectively,
“log(q
(−)
)”)
for
the
R-valued
function
on
M
ell
(Q),
hence
also
on
U
X
(Q),
obtained
by
forming
the
normalized
degree
“deg(−)”
of
the
effective
arithmetic
divisor
determined
by
the
q-parameters
of
an
elliptic
curve
over
a
number
field
at
arbitrary
nonarchimedean
primes
(respectively,
at
the
nonarchimedean
primes
that
do
not
divide
2)
[cf.
the
invariant
“log(q)”
as-
sociated,
in
the
statement
of
Theorem
1.10,
to
the
elliptic
curve
E
F
].
Also,
we
shall
write
ht
∞
for
the
[normalized]
height
function
on
U
X
(Q)
—
a
function
which
is
well-defined
up
to
a
“bounded
discrepancy”
[cf.
the
discussion
preced-
ing
[GenEll],
Proposition
3.4]
—
determined
by
the
pull-back
to
X
of
the
divisor
at
infinity
of
the
natural
compactification
(M
ell
)
Q
of
(M
ell
)
Q
.
Then
we
have
an
equality
of
“bounded
discrepancy
classes”
[cf.
[GenEll],
Definition
1.2,
(ii),
as
well
as
Remark
2.3.1,
(ii),
below]
2
1
6
·
log(q
(−)
)
≈
∀
1
6
·
log(q
(−)
)
≈
1
6
·
ht
∞
≈
ht
ω
X
(D)
of
functions
on
K
V
⊆
U
X
(Q).
(ii)
There
exist
·
a
positive
real
number
H
unif
which
is
independent
of
K
V
and
·
positive
real
numbers
C
K
and
H
K
which
depend
only
on
the
choice
of
the
compactly
bounded
subset
K
V
42
SHINICHI
MOCHIZUKI
such
that
the
following
property
is
satisfied:
Let
d
be
a
positive
integer,
d
a
positive
def
real
number
≤
1.
Set
δ
=
2
12
·
3
3
·
5
·
d.
Then
there
exists
a
finite
subset
Exc
d
⊆
U
X
(Q)
≤d
which
depends
only
on
K
V
,
d,
and
d
,
contains
all
points
corresponding
to
elliptic
curves
that
admit
automorphisms
of
order
>
2,
and
satisfies
the
following
property:
The
function
“log(q
∀
(−)
)”
of
(i)
is
4+
d
+
H
K
≤
H
unif
·
−3
d
·
d
on
Exc
d
.
Let
E
F
be
an
elliptic
curve
over
a
number
field
F
⊆
Q
that
determines
a
Q-valued
point
of
(M
ell
)
Q
which
lifts
[not
necessarily
uniquely!]
to
a
point
x
E
∈
U
X
(F
)
U
X
(Q)
≤d
such
that
x
E
∈
K
V
,
x
E
∈
Exc
d
.
Write
F
mod
for
the
minimal
field
of
definition
of
the
corresponding
point
∈
M
ell
(Q)
and
F
mod
⊆
def
F
tpd
=
F
mod
(
E
F
mod
[2]
)
⊆
F
for
the
“tripodal”
intermediate
field
obtained
from
F
mod
by
adjoining
the
fields
of
definition
of
the
2-torsion
points
of
any
model
of
E
F
×
F
Q
over
F
mod
[cf.
Proposition
1.8,
(ii),
(iii)].
Moreover,
we
assume
that
the
(3·5)-torsion
points
of
E
F
are
defined
over
F
,
and
that
F
=
F
mod
(
√
−1,
E
F
mod
[2
·
3
·
5]
)
def
=
F
tpd
(
√
−1,
E
F
tpd
[3
·
5]
)
√
—
i.e.,
that
F
is
obtained
from
F
tpd
by
adjoining
−1,
together
with
the
fields
of
definition
of
the
(3
·
5)-torsion
points
of
a
model
E
F
tpd
of
the
elliptic
curve
E
F
×
F
Q
over
F
tpd
determined
by
the
Legendre
form
of
the
Weierstrass
equation
discussed
above
[cf.
Proposition
1.8,
(vi)].
[Thus,
it
follows
from
Proposition
1.8,
(iv),
that
E
F
∼
=
E
F
tpd
×
F
tpd
F
over
F
,
so
x
E
∈
U
X
(F
tpd
)
⊆
U
X
(F
);
it
follows
from
Proposi-
tion
1.8,
(v),
that
E
F
has
stable
reduction
at
every
element
of
V(F
)
non
.]
Write
log(q
∀
)
(respectively,
log(q
2
))
for
the
result
of
applying
the
function
“log(q
∀
(−)
)”
2
(respectively,
“log(q
(−)
)”)
of
(i)
to
x
E
.
Then
E
F
and
F
mod
arise
as
the
“E
F
”
and
“F
mod
”
for
a
collection
of
initial
Θ-data
as
in
Theorem
1.10
that,
in
the
notation
of
Theorem
1.10,
satisfies
the
following
conditions:
(C1)
(log(q
∀
))
1/2
≤
l
≤
10δ
·
(log(q
∀
))
1/2
·
log(2δ
·
log(q
∀
));
(C2)
we
have
inequalities
1
6
·
log(q)
≤
2
1
6
·
log(q
)
≤
∀
1
6
·
log(q
)
≤
(1
+
E
)
·
(log-diff
X
(x
E
)
+
log-cond
D
(x
E
))
+
C
K
—
where
we
write
def
∀
)))
E
=
(60δ)
2
·
log(2δ·(log(q
(log(q
∀
))
1/2
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
43
[i.e.,
so
E
depends
on
the
integer
d,
as
well
as
on
the
elliptic
curve
E
F
!],
and
we
observe,
relative
to
the
notation
of
Theorem
1.10,
that
[it
follows
tautologically
from
the
definitions
that]
we
have
an
equality
log-diff
X
(x
E
)
=
log(d
F
tpd
),
as
well
as
inequalities
log(f
F
tpd
)
≤
log-cond
D
(x
E
)
≤
log(f
F
tpd
)
+
log(2l).
(iii)
The
positive
real
number
H
unif
of
(ii)
[which
is
independent
of
K
V
!]
may
be
chosen
in
such
a
way
that
the
following
property
is
satisfied:
Let
d
be
a
positive
integer,
d
and
positive
real
numbers
≤
1.
Then
there
exists
a
finite
subset
Exc
,d
⊆
U
X
(Q)
≤d
which
depends
only
on
K
V
,
,
d,
and
d
such
that,
in
the
notation
of
(ii),
the
function
“log(q
∀
(−)
)”
of
(i)
is
4+
d
≤
H
unif
·
−3
·
−3
+
H
K
d
·
d
on
Exc
,d
,
and,
moreover,
the
invariant
E
associated
to
an
elliptic
curve
E
F
as
in
(ii)
[i.e.,
that
satisfies
certain
conditions
which
depend
on
K
V
and
d]
satisfies
the
inequality
E
≤
whenever
the
point
x
E
∈
U
X
(F
)
satisfies
the
condition
x
E
∈
Exc
,d
.
Proof.
First,
we
consider
assertion
(i).
We
begin
by
observing
that,
in
light
of
the
condition
(∗
j-inv
)
that
was
imposed
on
K
V
,
it
follows
immediately
from
the
various
definitions
involved
that
2
log(q
(−)
)
≈
log(q
∀
(−)
)
—
where
we
observe
that
the
function
“log(q
∀
(−)
)”
may
be
identified
with
the
func-
tion
“deg
∞
”
of
the
discussion
preceding
[GenEll],
Proposition
3.4
—
on
K
V
⊆
U
X
(Q).
In
a
similar
vein,
since
the
support
of
K
V
contains
the
unique
archimedean
prime
of
Q,
it
follows
immediately
from
the
various
definitions
involved
[cf.
also
Remark
2.3.1,
(vi),
below]
that
log(q
∀
(−)
)
≈
ht
∞
on
K
V
⊆
U
X
(Q)
[cf.
the
argument
of
the
final
paragraph
of
the
proof
of
[GenEll],
2
Lemma
3.7].
Thus,
we
conclude
that
log(q
(−)
)
≈
log(q
∀
(−)
)
≈
ht
∞
on
K
V
⊆
U
X
(Q).
Finally,
since
[as
is
well-known]
the
pull-back
to
X
of
the
divisor
at
infinity
of
the
natural
compactification
(M
ell
)
Q
of
(M
ell
)
Q
is
of
degree
6,
while
the
line
bundle
ω
X
(D)
is
of
degree
1,
the
equality
of
BD-classes
16
·
ht
∞
≈
ht
ω
X
(D)
on
K
V
⊆
U
X
(Q)
follows
immediately
from
[GenEll],
Proposition
1.4,
(i),
(iii).
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
First,
let
us
recall
that
if
the
once-punctured
elliptic
curve
associated
to
E
F
fails
to
admit
an
F
-core,
then
there
are
only
four
possibilities
for
the
j-invariant
of
E
F
[cf.
[CanLift],
Proposition
2.7].
Thus,
if
we
take
the
set
Exc
d
to
be
the
[finite!]
collection
of
points
corresponding
to
these
four
j-invariants,
then
we
may
assume
that
the
once-punctured
elliptic
curve
associated
to
E
F
admits
an
F
-core,
hence,
in
particular,
does
not
have
any
automorphisms
of
order
>
2
over
Q.
In
the
discussion
to
follow,
it
will
be
necessary
to
enlarge
44
SHINICHI
MOCHIZUKI
the
finite
set
Exc
d
several
times,
always
in
a
fashion
that
depends
only
on
K
V
,
d,
and
d
[i.e.,
but
not
on
x
E
!]
and
in
such
a
way
that
the
function
“log(q
∀
(−)
)”
of
4+
d
+
H
K
on
Exc
d
for
some
positive
real
number
H
unif
that
(i)
is
≤
H
unif
·
−3
d
·
d
is
independent
of
K
V
and
some
positive
real
number
H
K
that
depends
only
on
K
V
[i.e.,
but
not
on
d
or
d
!].
Next,
let
us
write
h
=
log(q
∀
)
=
def
1
[F
:Q]
·
h
v
·
f
v
·
log(p
v
)
v∈V(F
)
non
—
that
is
to
say,
h
v
=
0
for
those
v
at
which
E
F
has
good
reduction;
h
v
∈
N
≥1
is
the
local
height
of
E
F
[cf.
[GenEll],
Definition
3.3]
for
those
v
at
which
E
F
has
bad
multiplicative
reduction.
Now
it
follows
[by
assertion
(i);
[GenEll],
Proposition
1.4,
(iv)]
that
the
inequality
h
1/2
<
ξ
prm
[cf.
the
notation
of
Proposition
2.1,
(ii)]
implies
that
there
is
only
a
finite
number
of
possibilities
for
the
j-invariant
of
E
F
.
Thus,
by
possibly
enlarging
the
finite
set
Exc
d
[in
a
fashion
that
depends
only
on
K
V
,
d,
and
d
and
in
such
a
way
that
h
≤
H
unif
on
Exc
d
for
some
positive
real
number
H
unif
that
is
independent
of
K
V
],
we
may
assume
without
loss
of
generality
that
the
inequality
h
1/2
≥
ξ
prm
≥
5
holds.
Thus,
since
[F
:
Q]
≤
δ
[cf.
the
properties
(E3),
(E4),
(E5)
in
the
proof
of
Theorem
1.10],
it
follows
that
h
−1/2
·
h
v
·
f
v
·
log(p
v
)
≥
h
−1/2
·
h
v
·
log(p
v
)
δ
·
h
1/2
≥
[F
:
Q]
·
h
1/2
=
≥
h
v
v
h
−1/2
·
h
v
·
log(p
v
)
≥
≥
h
1/2
h
v
v
log(p
v
)
≥
h
1/2
and
2δ
·
h
1/2
·
log(2δ
·
h)
≥
2
·
[F
:
Q]
·
h
1/2
·
log(2
·
[F
:
Q]
·
h)
≥
2
·
h
−1/2
·
log(2
·
h
v
·
f
v
·
log(p
v
))
·
h
v
·
f
v
·
log(p
v
)
h
v
=0
≥
h
−1/2
·
log(h
v
)
·
h
v
≥
h
v
=0
≥
h
v
h
−1/2
·
log(h
v
)
·
h
v
h
v
≥
h
1/2
log(h
v
)
≥
h
1/2
—
where
the
sums
are
all
over
v
∈
V(F
)
non
[possibly
subject
to
various
conditions,
as
indicated],
and
we
apply
the
elementary
estimate
2
·
log(p
v
)
≥
2
·
log(2)
=
log(4)
≥
1
[cf.
the
property
(E6)
in
the
proof
of
Theorem
1.10].
Thus,
in
summary,
we
conclude
from
the
estimates
made
above
that
if
we
take
A
to
be
the
[finite!]
set
of
prime
numbers
p
such
that
p
either
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
45
(S1)
is
≤
h
1/2
,
(S2)
divides
a
nonzero
h
v
for
some
v
∈
V(F
)
non
,
or
(S3)
is
equal
to
p
v
for
some
v
∈
V(F
)
non
for
which
h
v
≥
h
1/2
,
then
it
follows
from
Proposition
2.1,
(ii),
together
with
our
assumption
that
h
1/2
≥
ξ
prm
,
that,
in
the
notation
of
Proposition
2.1,
(ii),
θ
A
≤
2
·
h
1/2
+
δ
·
h
1/2
+
2δ
·
h
1/2
·
log(2δ
·
h)
≤
4δ
·
h
1/2
·
log(2δ
·
h)
≤
−ξ
prm
+
5δ
·
h
1/2
·
log(2δ
·
h)
—
where
we
apply
the
estimates
δ
≥
2
and
log(2δ
·
h)
≥
log(4)
≥
1
[cf.
the
property
(E6)
in
the
proof
of
Theorem
1.10].
In
particular,
it
follows
from
Proposition
2.1,
(i),
(ii),
together
with
our
assumption
that
h
1/2
≥
5
≥
1,
that
there
exists
a
prime
number
l
such
that
(P1)
(5
≤)
h
1/2
≤
l
≤
10δ
·
h
1/2
·
log(2δ
·
h)
(≤
20
·
δ
2
·
h
2
)
[cf.
the
condition
(C1)
in
the
statement
of
Corollary
2.2];
(P2)
l
does
not
divide
any
nonzero
h
v
for
v
∈
V(F
)
non
;
(P3)
if
l
=
p
v
for
some
v
∈
V(F
)
non
,
then
h
v
<
h
1/2
.
Next,
let
us
observe
that,
again
by
possibly
enlarging
the
finite
set
Exc
d
[in
a
fashion
that
depends
only
on
K
V
,
d,
and
d
and
in
such
a
way
that
h
≤
H
K
on
Exc
d
for
some
positive
real
number
H
K
that
depends
only
on
K
V
],
we
may
assume
without
loss
of
generality
that,
in
the
terminology
of
[GenEll],
Lemma
3.5,
(P4)
E
F
does
not
admit
an
l-cyclic
subgroup
scheme.
Indeed,
the
existence
of
an
l-cyclic
subgroup
scheme
of
E
F
would
imply
that
∀
l−2
24
·
log(q
)
≤
2
·
log(l)
+
T
K
—
where
we
apply
assertion
(i),
(P2),
the
displayed
inequality
of
[GenEll],
Lemma
3.5,
and
the
final
inequality
of
the
display
of
[GenEll],
Proposition
3.4;
we
take
the
“”
of
[GenEll],
Lemma
3.5,
to
be
1;
we
write
T
K
for
the
positive
real
number
[which
depends
only
on
the
choice
of
the
compactly
bounded
subset
K
V
]
that
results
from
the
various
“bounded
discrepancies”
implicit
in
these
inequalities.
Since
l
≥
5
[cf.
(P1)],
it
follows
that
1
≤
2
·
log(l)
≤
48
·
l−2
24
[cf.
the
property
(E6)
in
the
proof
of
Theorem
1.10],
and
hence
that
the
inequality
of
the
preceding
display
implies
that
log(q
∀
)
is
bounded.
On
the
other
hand,
[by
assertion
(i);
[GenEll],
Proposition
1.4,
(iv)]
this
implies
that
there
is
only
a
finite
number
of
possibilities
for
the
j-invariant
of
E
F
.
This
completes
the
proof
of
the
above
observation.
Next,
let
us
note
that
it
follows
immediately
from
(P1),
together
with
Propo-
sition
2.1,
(i),
that
h
1/2
·
log(l)
≤
h
1/2
·
log(20
·
δ
2
·
h
2
)
≤
2
·
h
1/2
·
log(5δ
·
h)
≤
8
·
h
1/2
·
log(2
·
δ
1/4
·
h
1/4
)
≤
8
·
h
1/2
·
2
·
δ
1/4
·
h
1/4
=
16
·
δ
1/4
·
h
3/4
46
SHINICHI
MOCHIZUKI
—
where
we
apply
the
estimates
20
≤
5
2
and
5
≤
2
4
.
In
particular,
we
observe
that,
again
by
possibly
enlarging
the
finite
set
Exc
d
[in
a
fashion
that
depends
only
on
K
V
,
d,
and
d
and
in
such
a
way
that
h
≤
H
unif
·d+H
K
on
Exc
d
for
some
positive
real
number
H
unif
that
is
independent
of
K
V
and
some
positive
real
number
H
K
that
depends
only
on
K
V
],
we
may
assume
without
loss
of
generality
that
def
(P5)
if
we
write
V
bad
mod
for
the
set
of
nonarchimedean
valuations
∈
V
mod
=
V(F
mod
)
that
do
not
divide
2l
and
at
which
E
F
has
bad
multiplicative
reduction,
then
V
bad
mod
=
∅.
Indeed,
if
V
bad
mod
=
∅,
then
it
follows,
in
light
of
the
definition
of
h,
from
(P3),
assertion
(i),
and
the
computation
performed
above,
that
h
≈
log(q
2
)
≤
h
1/2
·
log(l)
≤
16
·
δ
1/4
·
h
3/4
—
an
inequality
which
implies
that
h
1/4
,
hence
h
itself,
is
bounded.
On
the
other
hand,
[by
assertion
(i);
[GenEll],
Proposition
1.4,
(iv)]
this
implies
that
there
is
only
a
finite
number
of
possibilities
for
the
j-invariant
of
E
F
.
This
completes
the
proof
of
the
above
observation.
This
property
(P5)
implies
that
(P6)
the
image
of
the
outer
homomorphism
Gal(Q/F
)
→
GL
2
(F
l
)
determined
by
the
l-torsion
points
of
E
F
contains
the
subgroup
SL
2
(F
l
)
⊆
GL
2
(F
l
).
Indeed,
since,
by
(P5),
E
F
has
bad
multiplicative
reduction
at
some
valuation
∈
V
bad
mod
=
∅,
(P6)
follows
formally
from
(P2),
(P4),
and
[GenEll],
Lemma
3.1,
(iii)
[cf.
the
proof
of
the
final
portion
of
[GenEll],
Theorem
3.8].
Now
it
follows
formally
from
(P1),
(P2),
(P5),
and
(P6)
that,
if
one
takes
“F
”
to
be
Q,
“F
”
to
be
the
number
field
F
of
the
above
discussion,
“X
F
”
to
be
the
once-punctured
elliptic
curve
associated
to
E
F
,
“l”
to
be
the
prime
number
l
of
bad
the
above
discussion,
and
“V
bad
mod
”
to
be
the
set
V
mod
of
(P5),
then
there
exist
data
“C
K
”,
“V”,
and
“”
such
that
all
of
the
conditions
of
[IUTchI],
Definition
3.1,
(a),
(b),
(c),
(d),
(e),
(f
),
are
satisfied,
and,
moreover,
that
(P7)
the
resulting
initial
Θ-data
(F
/F,
X
F
,
l,
C
K
,
V,
V
bad
mod
,
)
satisfies
the
various
conditions
in
the
statement
of
Theorem
1.10.
Here,
we
note
in
passing
that
the
crucial
existence
of
data
“V”
and
“”
satisfying
the
requisite
conditions
follows,
in
essence,
as
a
consequence
of
the
fact
[i.e.,
(P6)]
that
the
Galois
action
on
l-torsion
points
contains
the
full
special
linear
group
SL
2
(F
l
).
In
light
of
(P7),
we
may
apply
Theorem
1.10
[cf.
also
Remark
1.10.7,
(i)]
to
conclude
that
1
6
·
log(q)
≤
(1
+
20·d
l
mod
)
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
20
·
(d
∗
mod
·
l
+
η
prm
)
≤
(1
+
δ
·
h
−1/2
)
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
200
·
δ
2
·
h
1/2
·
log(2δ
·
h)
+
20η
prm
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
47
—
where
we
apply
(P1),
as
well
as
the
estimates
20
·
d
mod
≤
d
∗
mod
≤
δ.
Next,
let
us
observe
that
it
follows
from
(P3),
together
with
the
computation
of
the
discussion
preceding
(P5),
that
2
1
6
·
log(q
)
−
16
·
log(q)
≤
1/2
1
·
log(l)
6
·
h
1/2
≤
h
≤
1/2
1
·
log(5δ
·
h)
3
·
h
·
log(2δ
·
h)
—
where
we
apply
the
estimates
1
≤
h
and
5
≤
2
3
.
Thus,
since,
by
assertion
(i),
the
difference
16
·log(q
∀
)−
16
·log(q
2
)
is
bounded
by
some
positive
real
number
B
K
[which
depends
only
on
the
choice
of
the
compactly
bounded
subset
K
V
],
we
conclude
that
1
6
·
h
=
∀
1
6
·
log(q
)
≤
(1
+
δ
·
h
−1/2
)
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
(15δ)
2
·
h
1/2
·
log(2δ
·
h)
+
12
·
C
K
≤
(1
+
δ
·
h
−1/2
)
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
16
·
h
·
25
·
(60δ)
2
·
h
−1/2
·
log(2δ
·
h)
+
12
·
C
K
def
—
where
we
write
C
K
=
40η
prm
+
2B
K
,
and
we
apply
the
estimate
6
·
5
≤
2
·
4
2
.
Now
let
us
set
def
E
=
(60δ)
2
·
h
−1/2
·
log(2δ
·
h)
(≥
5
·
δ
·
h
−1/2
);
def
∗
d
=
1
16
·
d
(<
12
≤
1)
—
where
we
apply
the
estimates
h
≥
1,
log(2δ
·
h)
≥
log(2δ)
≥
log(4)
≥
1
[cf.
the
property
(E6)
in
the
proof
of
Theorem
1.10],
and
d
≤
1.
Note
that
the
inequality
1
<
E
=
(60δ)
2
·
h
−1/2
·
log(2δ
·
h)
∗
∗
∗
=
(
∗
d
)
−1
·
(60δ)
2
·
h
−1/2
·
log(2
d
·
δ
d
·
h
d
)
∗
∗
≤
(
∗
d
)
−1
·
(60δ)
2+
d
·
h
−(1/2−
d
)
(1/2−
∗
d
)
∗
−3
4+
d
−1
≤
(
d
)
·
(60δ)
·
h
—
where
we
apply
Proposition
2.1,
(i),
together
with
the
estimates
1
16
=
≤
3;
1
∗
8
−
−
d
d
2
2
+
∗
d
32
+
d
=
≤
4
+
d
≤
5
1
∗
8
−
−
d
d
2
[both
of
which
are
consequences
of
the
fact
that
0
<
d
≤
1
≤
3],
as
well
as
the
estimates
0
<
∗
d
≤
1,
60δ
≥
2δ
≥
1,
and
h
≥
1
—
implies
a
bound
on
h,
hence,
[by
assertion
(i);
[GenEll],
Proposition
1.4,
(iv)]
that
there
is
only
a
finite
number
of
possibilities
for
the
j-invariant
of
E
F
.
Thus,
by
possibly
enlarging
the
finite
set
Exc
d
[in
a
fashion
that
depends
only
on
K
V
,
d,
and
d
and
in
such
a
way
that
4+
d
h
≤
H
unif
·
−3
+
H
K
on
Exc
d
for
some
positive
real
number
H
unif
that
is
d
·
d
independent
of
K
V
and
some
positive
real
number
H
K
that
depends
only
on
K
V
],
we
may
assume
without
loss
of
generality
that
E
≤
1.
48
SHINICHI
MOCHIZUKI
Thus,
in
summary,
we
obtain
inequalities
1
6
·
h
≤
(1
−
25
·
E
)
−1
(1
+
15
·
E
)
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
(1
−
25
·
E
)
−1
·
12
·
C
K
≤
(1
+
E
)
·
(log(d
F
tpd
)
+
log(f
F
tpd
))
+
C
K
by
applying
the
estimates
1
+
15
·
E
1
−
25
·
E
≤
1
+
E
;
1
−
25
·
E
≥
1
2
—
both
of
which
are
consequences
of
the
fact
that
0
<
E
≤
1.
Thus,
in
light
of
(P1),
together
with
the
observation
that
it
follows
immediately
from
the
definitions
[cf.
also
Proposition
1.8,
(vi)]
that
we
have
an
equality
log-diff
X
(x
E
)
=
log(d
F
tpd
),
as
well
as
inequalities
log(f
F
tpd
)
≤
log-cond
D
(x
E
)
≤
log(f
F
tpd
)+log(2l),
we
conclude
that
both
of
the
conditions
(C1),
(C2)
in
the
statement
of
assertion
(ii)
hold
for
C
K
as
defined
above.
This
completes
the
proof
of
assertion
(ii).
Finally,
assertion
(iii)
follows
immediately
by
applying
the
argument
applied
above
in
the
proof
of
assertion
(ii)
in
the
case
of
the
inequality
“1
<
E
”
to
the
inequality
“
<
E
”.
Remark
2.2.1.
(i)
Before
proceeding,
we
pause
to
examine
the
asymptotic
behavior
of
the
bound
obtained
in
Corollary
2.2,
(ii),
in
the
spirit
of
the
discussion
of
Remark
1.10.5,
(ii).
For
simplicity,
we
assume
that
F
tpd
=
Q
[so
d
mod
=
1];
we
write
def
def
h
=
log(q
∀
)
[cf.
the
proof
of
Corollary
2.2,
(ii)]
and
δ
=
log-diff
X
(x
E
)
+
log-cond
D
(x
E
)
=
log-cond
D
(x
E
)
[i.e.,
notation
that
is
closely
related
to
the
nota-
tion
of
Remark
1.10.5,
(ii),
but
differs
substantially
from
the
notation
of
Corollary
2.2,
(ii)].
Thus,
it
follows
immediately
from
the
definitions
that
1
<
log(3)
≤
δ
and
1
<
log(3)
≤
h.
In
particular,
the
bound
under
consideration
may
be
written
in
the
form
1/2
1
·
log(δ)
6
·
h
≤
δ
+
∗·
δ
—
where
“∗”
is
to
be
understood
as
denoting
a
fixed
positive
real
number;
we
observe
that
the
ratio
h/δ
is
always
a
positive
real
number
which
is
bounded
below
by
the
definition
of
h
and
δ
and
bounded
above
precisely
as
a
consequence
of
the
bound
under
consideration.
In
this
context,
it
is
of
interest
to
observe
that
the
form
of
the
“
term”
δ
1/2
·
log(δ)
is
strongly
reminiscent
of
well-known
interpretations
of
the
Riemann
hypothesis
in
terms
of
the
asymptotic
behavior
of
the
function
defined
by
considering
the
number
of
prime
numbers
less
than
a
given
natural
number.
Indeed,
from
the
point
of
view
of
weights
[cf.
also
the
discussion
of
Remark
2.2.2
below],
it
is
natural
to
regard
the
[logarithmic]
height
of
a
line
bundle
as
an
object
that
has
the
same
weight
as
a
single
Tate
twist,
or,
from
a
more
classical
point
of
view,
“2πi”
raised
to
the
power
1.
On
the
other
hand,
again
from
the
point
of
view
of
weights,
the
variable
“s”
of
the
Riemann
zeta
function
ζ(s)
may
be
thought
of
as
corresponding
precisely
to
the
number
of
Tate
twists
under
consideration,
so
a
single
Tate
twist
corresponds
to
“s
=
1”.
Thus,
from
this
point
of
view,
“s
=
12
”,
i.e.,
the
critical
line
that
appears
in
the
Riemann
hypothesis,
corresponds
precisely
to
the
square
roots
of
the
[logarithmic]
heights
under
consideration,
i.e.,
to
h
1/2
,
δ
1/2
.
Moreover,
from
the
point
of
view
of
the
computations
that
underlie
Theorem
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
49
1.10
and
Corollary
2.2,
(ii)
[cf.,
especially,
the
proof
of
Corollary
2.2,
(ii);
Steps
(v),
(viii)
of
the
proof
of
Theorem
1.10;
the
contribution
of
“b
i
”
in
the
second
displayed
inequality
of
Proposition
1.4,
(iii)],
this
δ
1/2
arises
as
a
result
of
a
sort
of
“balance”,
or
“duality”
—
i.e.,
that
occurs
as
one
increases
the
size
of
the
auxiliary
prime
l
[cf.
the
discussion
of
Remark
1.10.5,
(ii)]
—
between
the
archimedean
decrease
in
the
“
term”
δl
and
the
nonarchimedean
increase
in
the
“
term”
l
[i.e.,
that
arises
from
a
certain
estimate,
in
the
proof
of
Proposition
1.2,
(i),
(ii),
of
the
radius
of
convergence
of
the
p-adic
logarithm].
That
is
to
say,
such
a
global
arithmetic
duality
is
reminiscent
of
the
functional
equation
of
the
Riemann
zeta
function
[cf.
the
discussion
of
(iii)
below].
(ii)
In
[vFr],
§2,
it
is
conjectured
that,
in
the
notation
of
the
discussion
of
(i),
log
lim
sup
1
6
·h−δ
log(h)
=
1
2
and
observed
that
the
“
12
”
that
appears
here
is
strongly
reminiscent
of
the
“
12
”
that
appears
in
the
Riemann
hypothesis.
In
the
situation
of
Corollary
2.2,
(ii),
bounds
are
only
obtained
on
abc
sums
that
belong
to
the
compactly
bounded
subset
K
V
under
consideration;
such
bounds,
i.e.,
as
discussed
in
(i),
thus
imply
that
this
lim
sup
is
≤
12
.
On
the
other
hand,
it
is
shown
in
[vFr],
§2
[cf.
also
the
references
quoted
in
[vFr]],
that,
if
one
allows
arbitrary
abc
sums
[i.e.,
which
are
not
necessarily
assumed
to
be
contained
in
a
single
compactly
bounded
subset
K
V
],
then
this
lim
sup
is
≥
12
.
It
is
not
clear
to
the
author
at
the
time
of
writing
whether
or
not
such
estimates
[i.e.,
to
the
effect
that
the
lim
sup
under
consideration
is
≥
12
]
hold
even
if
one
imposes
the
restriction
that
the
abc
sums
under
consideration
be
contained
in
a
single
compactly
bounded
subset
K
V
.
(iii)
In
the
well-known
classical
theory
of
the
Riemann
zeta
function,
the
Riemann
zeta
function
is
closely
related
to
the
theta
function,
i.e.,
by
means
of
the
Mellin
transform.
In
light
of
the
central
role
played
by
theta
functions
in
the
theory
of
the
present
series
of
papers,
it
is
tempting
to
hope,
especially
in
the
context
of
the
observations
of
(i),
(ii),
that
perhaps
some
extension
of
the
theory
of
the
present
series
of
papers
—
i.e.,
some
sort
of
“inter-universal
Mellin
transform”
—
may
be
obtained
that
allows
one
to
relate
the
theory
of
the
present
series
of
papers
to
the
Riemann
zeta
function.
(iv)
In
the
context
of
the
discussion
of
(iii),
it
is
of
interest
to
recall
that,
rela-
tive
to
the
analogy
between
number
fields
and
one-dimensional
function
fields
over
finite
fields,
the
theory
of
the
present
series
of
papers
may
be
thought
of
as
being
analogous
to
the
theory
surrounding
the
derivative
of
a
lifting
of
the
Frobenius
morphism
[cf.
the
discussion
of
[IUTchI],
§I4;
[IUTchIII],
Remark
3.12.4].
On
the
other
hand,
the
analogue
of
the
Riemann
hypothesis
for
one-dimensional
func-
tion
fields
over
finite
fields
may
be
proven
by
considering
the
elementary
geometry
of
the
[graph
of
the]
Frobenius
morphism.
This
state
of
affairs
suggests
that
perhaps
some
sort
of
“integral”
of
the
theory
of
the
present
series
of
papers
could
shed
light
on
the
Riemann
hypothesis
in
the
case
of
number
fields.
(v)
One
way
to
summarize
the
point
of
view
discussed
in
(i),
(ii),
and
(iii)
is
as
follows:
The
asymptotic
behavior
discussed
in
(i)
suggests
that
perhaps
one
50
SHINICHI
MOCHIZUKI
should
expect
that
the
inequality
constituted
by
well-known
interpretations
of
the
Riemann
hypothesis
in
terms
of
the
asymptotic
behavior
of
the
function
defined
by
considering
the
number
of
prime
numbers
less
than
a
given
natural
number
may
be
obtained
as
some
sort
of
“restriction”
(ABC
inequality)|
canonical
number
of
some
sort
of
“ABC
inequality”
[i.e.,
some
sort
of
bound
of
the
sort
obtained
in
Corollary
2.2,
(ii)]
to
some
sort
of
“canonical
number”
[i.e.,
where
the
term
“number”
is
to
be
understood
as
referring
to
an
abc
sum].
Here,
the
descriptive
“canonical”
is
to
be
understood
as
expressing
the
idea
that
one
is
not
so
much
interested
in
considering
a
fixed
explicit
“number/abc
sum”,
but
rather
some
sort
of
suitable
abstraction
of
the
sort
of
sequence
of
numbers/abc
sums
that
gives
rise
to
the
lim
sup
value
of
“
12
”
discussed
in
(ii).
Of
course,
it
is
by
no
means
clear
precisely
how
such
an
“abstraction”
should
be
formulated,
but
the
idea
is
that
it
should
represent
some
sort
of
average
over
all
possible
addition
operations
in
the
number
field
[in
this
case,
Q]
under
consideration
or
[perhaps
equivalently]
some
sort
of
“arithmetic
measure
or
distribution”
constituted
by
such
a
collection
of
all
possible
addition
operations
that
somehow
amounts
to
a
sort
of
arithmetic
analogue
of
the
measure
that
gives
rise
to
the
classical
Mellin
transform
[i.e.,
that
appears
in
the
discussion
of
(iii)].
Remark
2.2.2.
In
the
context
of
the
discussion
of
weights
in
Remark
2.2.1,
(i),
it
is
of
interest
to
recall
the
significance
of
the
Gaussian
integral
∞
e
−x
dx
=
2
√
π
−∞
in
the
theory
of
the
present
series
of
papers
[cf.
[IUTchII],
Introduction;
[IUTchII],
Remark
1.12.5,
as
well
as
Remark
1.10.1
of
the
present
paper].
Indeed,
typically
discussions
of
the
Riemann
zeta
function
ζ(s),
or
more
general
L-functions,
in
the
context
of
conventional
arithmetic
geometry
are
concerned
principally
with
the
behavior
of
such
functions
at
integral
values
[i.e.,
∈
Z]
of
the
variable
s.
Such
integral
values
of
the
variable
s
correspond
to
integral
Tate
twists,
i.e.,
at
a
more
concrete
level,
to
integral
powers
of
the
quantity
2πi.
If
one
neglects
nonzero
factors
∈
Q(i),
then
such
integral
powers
may
be
regarded
as
integral
powers
of
π
[or
2π].
At
the
level
of
classical
integrals,
the
notion
of
a
single
Tate
twist
may
be
thought
of
as
corresponding
to
the
integral
dθ
=
2π
S
1
over
the
unit
circle
S
1
;
at
the
level
of
schemes,
the
notion
of
a
single
Tate
twist
may
be
thought
of
as
corresponding
to
the
scheme
G
m
.
On
the
other
hand,
whereas
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
51
the
conventional
theory
of
Tate
twists
in
arithmetic
geometry
only
involves
integral
powers
of
a
single
Tate
twist,
i.e.,
corresponding,
in
essence,
to
integral
powers
of
π,
the
Gaussian
integral
may
be
thought
of
as
a
sort
of
fundamental
integral
representation
of
the
notion
of
a
“Tate
semi-twist”.
From
this
point
of
view,
scheme-theoretic
Hodge-Arakelov
theory
may
be
thought
of
as
a
sort
of
fundamen-
tal
scheme-theoretic
represention
of
the
notion
of
a
“Tate
semi-twist”
[cf.
the
discussion
of
[IUTchII],
Remark
1.12.5].
Thus,
in
summary,
(a)
the
Gaussian
integral,
(b)
scheme-theoretic
Hodge-Arakelov
theory,
(c)
the
inter-universal
Teichmüller
theory
developed
in
the
present
series
of
papers,
and
(d)
the
Riemann
hypothesis,
may
all
be
thought
of
as
“phenomena
of
weight
12
”,
i.e.,
at
a
concrete
level,
√
phenomena
that
revolve
around
arithmetic
versions
of
“
π”.
Moreover,
we
observe
that
in
the
first
three
of
these
four
examples,
the
essential
nature
of
the
notion
of
“weight
12
”
may
be
thought
of
as
being
reflected
in
some
sort
of
exponential
of
a
quadratic
form.
This
state
of
affairs
is
strongly
reminiscent
of
(1)
the
Griffiths
semi-transversality
of
the
crystalline
theta
object
that
occurs
in
scheme-theoretic
Hodge-Arakelov
theory
[cf.
[HASurII],
Theorem
2.8;
[IUTchII],
Remark
1.12.5,
(i)],
which
corresponds
essentially
[cf.
the
discussion
of
the
proof
of
[HASurII],
Theorem
2.10]
to
the
quadratic
form
that
appears
in
the
exponents
of
the
well-known
series
expansion
of
the
theta
function;
(2)
the
quadratic
nature
of
the
commutator
of
the
theta
group,
which
is
applied,
in
[EtTh]
[cf.
the
discussion
of
[IUTchIII],
Remark
2.1.1],
to
derive
the
various
rigidity
properties
which
are
interpreted,
in
[IUTchII],
§1,
as
multiradiality
properties
—
an
interpretation
that
is
strongly
rem-
iniscent,
if
one
interprets
“multiradiality”
in
terms
of
“connections”
and
“parallel
transport”
[cf.
[IUTchII],
Remark
1.7.1],
of
the
quadratic
form
discussed
in
(1);
(3)
the
essentially
quadratic
nature
of
the
“
term”
∗
·
δl
+
∗
·
l
[which,
we
recall,
occurs
at
the
level
of
addition
of
heights,
i.e.,
log-volumes!]
in
the
discussion
of
Remark
1.10.5,
(ii).
Remark
2.2.3.
The
discussion
of
Remark
2.2.1
centers
around
the
content
of
Corollary
2.2,
(ii),
in
the
case
of
elliptic
curves
defined
over
Q.
On
the
other
hand,
if,
in
the
context
of
Corollary
2.2,
(ii),
(iii),
one
considers
the
case
where
d
is
an
arbitrary
positive
integer
[i.e.,
which
is
not
necessarily
bounded,
as
in
the
situation
of
Corollary
2.3
below!],
then
the
inequalities
obtained
in
(C2)
of
Corollary
2.2,
(ii),
may
be
regarded,
by
applying
Corollary
2.2,
(iii),
as
a
sort
of
“weak
version”
of
the
so-called
“uniform
ABC
Conjecture”.
That
is
to
say,
these
inequalities
constitute
only
a
“weak
version”
in
the
sense
that
they
are
restricted
to
rational
points
that
lie
in
the
compactly
bounded
subset
K
V
,
and,
moreover,
the
bounds
52
SHINICHI
MOCHIZUKI
given
for
the
function
“log(q
∀
(−)
)”
[i.e.,
in
essence,
the
“height”]
on
Exc
d
and
Exc
,d
depend
on
the
positive
integer
d
[cf.
also
Remark
2.3.2,
(i),
below].
Remark
2.2.4.
Before
proceeding,
it
is
perhaps
of
interest
to
consider
the
ideas
discussed
in
Remarks
2.2.1,
2.2.3
above
in
the
context
of
the
analogy
between
the
theory
of
the
present
series
of
papers
and
the
p-adic
Teichmüller
theory
of
[pOrd],
[pTeich]
[cf.
also
[InpTch]].
(i)
The
analogy
between
the
theory
of
the
present
series
of
papers
and
the
p-
adic
Teichmüller
theory
of
[pOrd],
[pTeich]
[cf.
also
[InpTch]]
is
discussed
in
detail
in
[IUTchIII],
Remark
1.4.1,
(iii);
[IUTchIII],
Remark
3.12.4.
In
a
word,
this
discussion
concerns
similarities
between
the
log-theta-lattice
considered
in
the
present
series
of
papers
and
the
canonical
Frobenius
lifting
on
the
ordinary
locus
of
a
canonical
curve
of
the
sort
that
appears
in
the
theory
of
[pOrd].
Such
a
canonical
curve
is
associated,
in
the
theory
of
[pOrd],
to
a
hyperbolic
curve
equipped
with
a
nilpotent
ordinary
indigenous
bundle
over
a
perfect
field
of
positive
characteristic
p.
On
the
other
hand,
the
theory
of
[pOrd]
also
addresses
the
universal
case,
i.e.,
of
the
tautological
hyperbolic
curve
equipped
with
a
nilpotent
ordinary
indigenous
bundle
over
the
moduli
stack
of
such
data
in
positive
characteristic.
In
particular,
one
constructs,
in
the
theory
of
[pOrd],
a
canonical
Frobenius
lifting
over
a
canonical
p-adic
lifting
of
this
moduli
stack.
This
moduli
stack
is
smooth
of
dimension
3g
−
3
+
r
[i.e.,
in
the
case
of
hyperbolic
curves
of
type
(g,
r)]
over
F
p
,
hence,
in
particular,
is
far
from
perfect
[i.e.,
as
an
algebraic
stack
in
positive
characteristic].
Thus,
in
some
sense,
the
gap
between
the
theory
of
the
present
series
of
papers,
on
the
one
hand,
and
the
notion
discussed
in
Remark
2.2.1,
(v),
of
a
“canonical
number/arithmetic
measure/distribution”,
on
the
other,
may
be
understood,
in
the
context
of
the
analogy
with
p-adic
Teichmüller
theory,
as
corresponding
to
the
gap
between
the
theory
of
[pOrd]
specialized
to
the
case
of
“canonical
curves”,
i.e.,
over
perfect
base
fields,
and
the
full,
non-
specialized
version
of
the
theory
of
[pOrd],
i.e.,
which
concerns
canonical
Frobenius
liftings
over
the
non-perfect
moduli
stack
of
hyperbolic
curves
equipped
with
a
nilpotent
ordinary
indigenous
bundle.
That
is
to
say,
in
a
word,
one
has
a
correspondence
“canonical
number”
←→
modular
Frobenius
liftings.
(ii)
In
general,
the
gap
between
perfect
and
non-perfect
schemes
in
positive
characteristic
is
reflected
precisely
in
the
extent
to
which
the
Frobenius
morphism
on
the
scheme
under
consideration
fails
to
be
an
isomorphism.
Put
another
way,
the
“phenomenon”
of
non-perfect
schemes
in
positive
characteristic
may
be
thought
of
as
a
reflection
of
the
distortion
arising
from
the
Frobenius
morphism
in
positive
characteristic.
In
the
context
of
the
theory
of
the
present
series
of
papers
[cf.
[IUTchIII],
Remark
1.4.1,
(iii)],
the
Frobenius
morphism
in
positive
characteristic
corresponds
to
the
log-link.
Moreover,
in
the
context
of
the
inequalities
obtained
in
Theorem
1.10,
the
term
“∗
·
l”
[cf.
the
discussion
of
Remark
1.10.5,
(ii)]
arises,
in
the
computations
that
underlie
the
proof
of
Theorem
1.10,
precisely
by
applying
the
prime
number
theorem
[i.e.,
Proposition
1.6]
to
sum
up
the
log-volumes
of
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
53
log-shells
[cf.
Propositions
1.2,
(ii);
1.4,
(iii)]
at
various
nonarchimedean
primes
of
the
number
field.
In
this
context,
we
make
the
following
observations:
·
These
log-volumes
of
log-shells
may
be
thought
of
as
numerical
measures
of
the
distortions
of
the
integral
structure
[i.e.,
relative
to
the
“arithmetic
holomorphic”
integral
structures
determined
by
the
various
local
rings
of
integers
“O”]
that
arise
from
the
log-link.
·
Estimates
arising
from
the
prime
number
theorem
are
closely
related
to
the
aspects
of
the
Riemann
zeta
function
that
are
discussed
in
Remark
2.2.1.
·
The
prime
number
l
is,
ultimately,
in
the
computations
of
Corollary
2.2,
(ii)
[cf.,
especially,
condition
“(C1)”],
taken
to
be
roughly
of
the
order
of
the
square
root
of
the
height
of
the
elliptic
curve
under
consideration.
That
is
to
say,
since
the
height
of
an
elliptic
curve
“roughly
controls”
[i.e.,
up
to
finitely
many
possibilities]
the
moduli
of
the
elliptic
curve,
the
prime
number
l
may
be
thought
of
as
a
sort
of
rough
numerical
representation
of
the
moduli
of
the
elliptic
curve
under
consideration.
Thus,
in
summary,
these
observations
strongly
support
the
point
of
view
that
the
computations
that
underlie
the
proof
of
Theorem
1.10
may
be
thought
of
as
constituting
one
convincing
piece
of
evidence
for
the
point
of
view
discussed
in
(i)
above.
(iii)
In
the
context
of
the
discussion
of
(i),
(ii),
it
is
of
interest
to
recall
that
the
modular
Frobenius
liftings
of
[pOrd]
are
not
defined
over
the
algebraic
moduli
stack
of
hyperbolic
curves
over
Z
p
,
but
rather
over
the
p-adic
formal
algebraic
stack
[which
is
formally
étale
over
the
corresponding
algebraic
moduli
stack
of
hyperbolic
curves
over
Z
p
]
constituted
by
the
canonical
lifting
to
Z
p
of
the
moduli
stack
of
hyperbolic
curves
equipped
with
a
nilpotent
ordinary
indigenous
bundle.
That
is
to
say,
the
gap
between
this
[“p-adically
analytic”]
p-adic
formal
algebraic
stack
parametrizing
“ordinary”
data
and
the
corresponding
algebraic
moduli
stack
of
hyperbolic
curves
over
Z
p
is
highly
reminiscent,
in
the
context
of
Corollary
2.2,
(ii)
[cf.
also
Remark
2.2.3],
of
the
gap
between
the
[“arith-
metically
analytic”]
compactly
bounded
subsets
“K
V
”
[i.e.,
consisting
of
elliptic
curves
that
satisfy
the
condition
of
being
in
“sufficiently
gen-
eral
position”
—
a
condition
that
may
be
thought
of
as
a
sort
of
“global
arithmetic
version
of
ordinariness”]
and
the
entire
set
of
algebraic
points
“M
ell
(Q)”.
Ultimately,
this
gap
between
“K
V
”
and
“M
ell
(Q)”
will
be
bridged,
in
Corollary
2.3
below,
by
applying
[GenEll],
Theorem
2.1,
which
may
be
thought
of
as
a
sort
of
arithmetic
analytic
continuation
by
means
of
[noncritical]
Belyi
maps
[cf.
the
discussion
of
Belyi
maps
in
the
Introduction
to
[GenEll]].
This
state
of
affairs
is
reminiscent
of
the
“arithmetic
analytic
continuation
via
Belyi
maps”
that
occurs
in
the
theory
of
[AbsTopIII]
[i.e.,
in
essence,
the
theory
of
Belyi
cuspidalizations]
that
is
applied
in
[IUTchI],
§5
[cf.
[IUTchI],
Remark
5.1.4].
Finally,
in
this
context,
54
SHINICHI
MOCHIZUKI
we
recall
that
the
open
immersion
“
κ
”
that
appears
in
the
discussion
towards
the
end
of
[InpTch],
§2.6
—
i.e.,
which
embeds
a
sort
of
perfection
of
the
p-adic
formal
algebraic
stack
discussed
above
into
an
essentially
algebraic
stack
given
by
a
certain
pro-finite
covering
of
the
corresponding
algebraic
moduli
stack
of
hyperbolic
curves
over
Z
p
determined
by
considering
representations
of
the
geometric
fundamental
group
into
P
GL
2
(Z
p
)
—
may
be
thought
of
as
a
sort
of
p-adic
analytic
continuation
to
this
corresponding
algebraic
moduli
stack
of
the
essentially
“p-adically
analytic”
theory
of
modular
Frobenius
liftings
developed
in
[pOrd].
(iv)
Finally,
in
the
context
of
the
discussion
of
(i),
(ii),
(iii),
we
observe
that
the
issue
discussed
in
Remark
2.2.1
of
considering
the
asymptotic
behavior
of
the
theory
of
the
present
series
of
papers
when
l
→
∞
may
be
thought
of
as
the
problem
of
understanding
how
the
theory
of
the
present
series
of
papers
behaves
as
one
passes
from
the
discrete
approximation
of
the
elliptic
curve
under
consideration
constituted
by
the
l-torsion
points
of
the
elliptic
curve
to
the
“full
continuous
theory”
[cf.
the
discussion
of
[IUTchI],
Remark
6.12.3,
(i)
;
[HASurI],
§1.3.4].
This
point
of
view
is
of
interest
in
light
of
the
theory
of
Bernoulli
numbers,
i.e.,
which,
on
the
one
hand,
is,
as
is
well-known,
closely
related
to
the
values
[at
positive
even
integers]
of
the
Riemann
zeta
function
[cf.
the
discussion
of
Remark
2.2.1],
and,
on
the
other
hand,
is
closely
related
to
the
passage
from
the
discrete
difference
operator
f
(x)
→
f
(x
+
1)
−
f
(x)
—
for,
say,
real-valued
real
analytic
functions
f
(−)
on
the
real
line
—
to
the
d
f
(x)
continuous
derivative
operator
f
(x)
→
dx
—
where
we
recall
that
the
operator
f
(x)
→
f
(x
+
1)
may
be
thought
of
as
the
d
operator
“e
dx
”
obtained
by
exponentiating
this
continuous
derivative
operator.
We
are
now
ready
to
state
and
prove
the
main
theorem
of
the
present
§2,
which
may
also
be
regarded
as
the
main
application
of
the
theory
developed
in
the
present
series
of
papers.
Corollary
2.3.
(Diophantine
Inequalities)
Let
X
be
a
smooth,
proper,
def
geometrically
connected
curve
over
a
number
field;
D
⊆
X
a
reduced
divisor;
U
X
=
X\D;
d
a
positive
integer;
∈
R
>0
a
positive
real
number.
Write
ω
X
for
the
canonical
sheaf
on
X.
Suppose
that
U
X
is
a
hyperbolic
curve,
i.e.,
that
the
degree
of
the
line
bundle
ω
X
(D)
is
positive.
Then,
relative
to
the
notation
reviewed
above,
one
has
an
inequality
of
“bounded
discrepancy
classes”
ht
ω
X
(D)
(1
+
)(log-diff
X
+
log-cond
D
)
of
functions
on
U
X
(Q)
≤d
—
i.e.,
the
function
(1
+
)(log-diff
X
+
log-cond
D
)
−
ht
ω
X
(D)
is
bounded
below
by
a
constant
on
U
X
(Q)
≤d
[cf.
[GenEll],
Definition
1.2,
(ii),
as
well
as
Remark
2.3.1,
(ii),
below].
Proof.
One
verifies
immediately
that
the
content
of
the
statement
of
Corollary
2.3
coincides
precisely
with
the
content
of
[GenEll],
Theorem
2.1,
(i).
Thus,
it
follows
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
55
from
the
equivalence
of
[GenEll],
Theorem
2.1,
that,
in
order
to
complete
the
proof
of
Corollary
2.3,
it
suffices
to
verify
that
[GenEll],
Theorem
2.1,
(ii),
holds.
That
is
to
say,
we
may
assume
without
loss
of
generality
that:
·
X
=
P
1
Q
is
the
projective
line
over
Q;
·
D
⊆
X
is
the
divisor
consisting
of
the
three
points
“0”,
“1”,
and
“∞”;
·
K
V
⊆
U
X
(Q)
is
a
compactly
bounded
subset
[cf.
Remark
2.3.1,
(vi),
below]
whose
support
contains
the
nonarchimedean
prime
“2”;
·
K
V
satisfies
the
condition
“(∗
j-inv
)”
of
Corollary
2.2.
[Here,
we
note,
with
regard
to
the
condition
“(∗
j-inv
)”
of
Corollary
2.2,
that
this
condition
only
concerns
the
behavior
of
K
V
U
X
(Q)
≤d
as
d
varies;
that
is
to
say,
this
condition
is
entirely
vacuous
in
situations,
i.e.,
such
as
the
situation
considered
in
[GenEll],
Theorem
2.1,
(ii),
in
which
one
is
only
concerned
with
K
V
U
X
(Q)
≤d
for
a
fixed
d.]
Then
it
suffices
to
show
that
the
inequality
of
BD-classes
of
functions
[cf.
[GenEll],
Definition
1.2,
(ii),
as
well
as
Remark
2.3.1,
(ii),
below]
ht
ω
X
(D)
(1
+
)(log-diff
X
+
log-cond
D
)
holds
on
K
V
U
X
(Q)
≤d
.
But
such
an
inequality
follows
immediately,
in
light
of
the
[relevant]
equality
of
BD-classes
of
Corollary
2.2,
(i),
from
Corollary
2.2,
(ii)
[cf.
condition
(C2)],
(iii)
[where
we
note
that
it
follows
immediately
from
the
various
definitions
involved
that
d
mod
≤
d].
This
completes
the
proof
of
Corollary
2.3.
Remark
2.3.1.
We
take
this
opportunity
to
correct
some
unfortunate
misprints
in
[GenEll].
(i)
The
notation
“ord
v
(−)
:
F
v
→
Z”
in
the
final
sentence
of
the
first
paragraph
following
[GenEll],
Definition
1.1,
should
read
“ord
v
(−)
:
F
v
×
→
Z”.
(ii)
In
[GenEll],
Definition
1.2,
(ii),
the
non-resp’d
and
first
resp’d
items
in
the
display
should
be
reversed!
That
is
to
say,
the
notation
“α
F
β”
corresponds
to
“α(x)
−
β(x)
≤
C”;
the
notation
“α
F
β”
corresponds
to
“β(x)
−
α(x)
≤
C”.
(iii)
The
first
portion
of
the
first
sentence
of
the
statement
of
[GenEll],
Corollary
4.4,
should
read:
“Let
Q
be
an
algebraic
closure
of
Q;
.
.
.
”.
(iv)
The
“log-diff
M
ell
([E
L
]))”
in
the
second
inequality
of
the
final
display
of
the
statement
of
[GenEll],
Corollary
4.4,
should
read
“log-diff
M
ell
([E
L
])”.
(v)
The
equality
ht
E
≈
(deg(E)/deg(ω
X
))
·
ht
ω
X
implicit
in
the
final
“≈”
of
the
final
display
of
the
proof
of
[GenEll],
Theorem
2.1,
should
be
replaced
by
an
inequality
ht
E
2
·
(deg(E)/deg(ω
X
))
·
ht
ω
X
[which
follows
immediately
from
[GenEll],
Proposition
1.4,
(ii)],
and
the
expression
“deg(E)/deg(ω
X
)”
in
the
inequality
imposed
on
the
choice
of
should
be
replaced
by
the
expression
“2
·
(deg(E)/deg(ω
X
))”.
56
SHINICHI
MOCHIZUKI
(vi)
Suppose
that
we
are
in
the
situation
of
[GenEll],
Example
1.3,
(ii).
Let
U
⊆
X
be
an
open
subscheme.
Then
a
“compactly
bounded
subset”
K
V
⊆
U
(Q)
(⊆
X(Q))
of
U
(Q)
is
to
be
understood
as
a
subset
which
forms
a
compactly
bounded
subset
of
X(Q)
[i.e.,
in
the
sense
discussed
in
[GenEll],
Example
1.3,
(ii)]
and,
moreover,
def
satisfies
the
property
that
for
each
v
∈
V
arc
=
V
V(Q)
arc
(respectively,
v
∈
def
V
non
=
V
V(Q)
non
),
the
compact
domain
K
v
⊆
X
arc
(respectively,
K
v
⊆
X(Q
v
))
is,
in
fact,
contained
in
U
(C)
⊆
X(C)
=
X
arc
(respectively,
U
(Q
v
)
⊆
X(Q
v
)).
In
particular,
this
convention
should
be
applied
to
the
use
of
the
term
“compactly
bounded
subset”
in
the
statements
of
[GenEll],
Theorem
2.1;
[GenEll],
Lemma
3.7;
[GenEll],
Theorem
3.8;
[GenEll],
Corollary
4.4,
as
well
as
in
the
present
paper
[cf.
the
statement
of
Corollary
2.2;
the
proof
of
Corollary
2.3].
Although
this
convention
was
not
discussed
explicitly
in
[GenEll],
Example
1.3,
(ii),
it
is,
in
effect,
discussed
explicitly
in
the
discussion
of
“compactly
bounded
subsets”
at
the
beginning
of
the
Introduction
to
[GenEll].
Moreover,
this
convention
is
implicit
in
the
arguments
involving
compactly
bounded
subsets
in
the
proof
of
[GenEll],
Theorem
2.1.
(vii)
In
the
discussion
following
the
second
display
of
[GenEll],
Example
1.3,
(ii),
the
phrase
“(respectively,
X(Q
v
))”
should
read
“(respectively,
X(Q
v
))”.
(viii)
The
first
display
of
the
paragraph
immediately
following
[GenEll],
Re-
mark
3.3.1,
should
read
as
follows:
2
def
|α|
=
α
∧
α
E
v
[i.e.,
the
integral
should
be
replaced
by
the
absolute
value
of
the
integral].
Remark
2.3.2.
(i)
The
reader
will
note
that,
by
arguing
with
a
“bit
more
care”,
it
is
not
difficult
to
give
stronger
versions
of
the
various
estimates
that
occur
in
Theorem
1.10;
Corollaries
2.2,
2.3
and
their
proofs.
Such
stronger
estimates
are,
however,
beyond
the
scope
of
the
present
series
of
papers,
so
we
shall
not
pursue
this
topic
further
in
the
present
paper.
(ii)
On
the
other
hand,
we
observe
that
the
constant
“1”
in
the
inequality
of
the
display
of
Corollary
2.3
cannot
be
improved
—
cf.
the
examples
constructed
in
[Mss];
the
discussion
of
Remark
1.10.5,
(ii),
(iii).
This
observation
is
closely
related
to
discussions
of
how
the
theory
of
the
present
series
of
papers
breaks
down
if
one
attempts
to
replace
the
first
power
of
the
étale
theta
function
by
its
N
-th
power
for
some
integer
N
≥
2
[cf.
the
discussion
in
the
final
portion
of
Step
(xi)
of
the
proof
of
[IUTchIII],
Corollary
3.12;
the
discussion
of
[IUTchIII],
Remark
3.12.1,
(ii)].
Such
an
“N
-th
power
operation”
may
also
be
thought
of
as
corresponding
to
the
operation
of
replacing
each
Tate
curve
that
occurs
at
an
element
∈
V
bad
by
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
57
the
Tate
curve
whose
q-parameter
is
given
by
the
N
-th
power
of
the
q-parameter
of
the
original
Tate
curve.
This
sort
of
operation
on
Tate
curves
may,
in
turn,
be
thought
of
as
an
isogeny
of
the
sort
that
occurs
in
[GenEll],
Lemma
3.5.
On
the
other
hand,
the
content
of
the
proof
of
[GenEll],
Lemma
3.5,
consists
essentially
of
a
computation
to
the
effect
that
even
if
one
attempts
to
consider
such
“N
-th
power
isogenies”
at
certain
elements
∈
V
bad
,
the
global
height
of
the
elliptic
curve
over
a
number
field
that
arises
from
such
an
isogeny
will
typically
remain,
up
to
a
relatively
small
discrepancy,
unchanged.
In
this
context,
we
recall
that
this
sort
of
invariance,
up
to
a
relatively
small
discrepancy,
of
the
global
height
under
isogeny
is
one
of
the
essential
observations
that
underlies
the
theory
of
[Falt]
—
a
state
of
affairs
that
is
also
of
interest
in
light
of
the
observations
of
Remark
2.3.3
below.
Remark
2.3.3.
Corollary
2.3
may
be
thought
of
as
an
effective
version
of
the
Mordell
Conjecture.
From
this
point
of
view,
it
is
perhaps
of
interest
to
compare
the
“essential
ingredients”
that
are
applied
in
the
proof
of
Corollary
2.3
[i.e.,
in
effect,
that
are
applied
in
the
present
series
of
papers!]
with
the
“essential
ingredients”
applied
in
[Falt].
The
following
discussion
benefited
substantially
from
numerous
e-mail
and
skype
exchanges
with
Ivan
Fesenko
during
the
summer
of
2015.
(i)
Although
the
author
does
not
wish
to
make
any
pretensions
to
completeness
in
any
rigorous
sense,
perhaps
a
rough,
informal
list
of
“essential
ingredients”
in
the
case
of
[Falt]
may
be
given
as
follows:
(a)
results
in
elementary
algebraic
number
theory
related
to
the
“geometry
of
numbers”,
such
as
the
theory
of
heights
and
the
Hermite-Minkowski
theorem;
(b)
the
global
class
field
theory
of
number
fields;
(c)
the
p-adic
theory
of
Hodge-Tate
decompositions;
(d)
the
p-adic
theory
of
finite
flat
group
schemes;
(e)
generalities
in
algebraic
geometry
concerning
isogenies
and
Tate
modules
of
abelian
varieties;
(f)
generalities
in
algebraic
geometry
concerning
polarizations
of
abelian
varieties;
(g)
the
logarithmic
geometry
of
toroidal
compactifications
of
the
moduli
stack
of
abelian
varieties.
With
regard
to
the
global
class
field
theory
of
(b),
we
observe
that
there
are
nu-
merous
different
approaches
to
“dissecting”
the
proofs
of
the
main
results
of
global
class
field
theory
into
more
primitive
components.
To
some
extent,
these
different
approaches
correspond
to
different
points
of
view
arising
from
subsequent
research
on
topics
related
to
global
class
field
theory.
Here,
we
wish
to
consider
the
approach
taken
in
[Lang1],
Chapters
VIII,
IX,
X,
XI,
which
is
attibuted
[cf.
the
Introduction
to
[Lang1],
Part
Two]
to
Weber.
It
is
of
interest,
in
the
context
of
the
discussion
of
(vii)
below,
that
this
is
apparently
the
oldest
approach
to
proving
certain
por-
tions
of
global
class
field
theory.
It
is
also
of
interest
that
this
approach
motivates
the
approach
to
global
class
field
theory
via
consideration
of
density
of
primes
in
arithmetic
progressions
and
splitting
laws.
This
aspect
of
this
approach
of
[Lang1]
is
closely
related
to
various
issues
that
appear
in
[Falt]
[cf.
[Lang1],
Chapter
VIII,
58
SHINICHI
MOCHIZUKI
§5].
Moreover,
as
we
shall
see
in
the
following
discussion,
this
approach
of
[Lang1]
to
global
class
field
theory
is
well-suited
to
discussions
of
comparisons
between
the
theory
of
[Falt]
and
the
inter-universal
Teichmüller
theory
developed
in
the
present
series
of
papers.
At
a
technical
level,
the
dissection
of
the
global
class
field
theory
of
(b),
as
developed
in
[Lang1],
into
more
primitive
components
may
be
summarized
as
follows:
(b-1)
the
local
class
field
theory
of
p-adic
local
fields
[cf.
[Lang1],
Chapter
IX,
§3;
[Lang1],
Chapter
XI,
§4];
(b-2)
the
theory
of
global
density
of
primes
[cf.
the
discussion
surrounding
the
Universal
Norm
Index
Inequality
in
[Lang1],
Chapter
VIII,
§3];
(b-3)
results
in
elementary
algebraic
number
theory
related
to
the
“geometry
of
numbers”
that
give
rise
to
the
Unit
Theorem
[cf.
[Lang1],
Chapter
V,
§1;
[Lang1],
Chapter
IX,
§4];
(b-4)
the
global
reciprocity
law,
i.e.,
in
effect,
the
existence
of
a
conductor
for
the
Artin
symbol
[cf.
[Lang1],
Chapter
X,
§2];
(b-5)
Kummer
theory
[cf.
[Lang1],
Chapter
XI,
§1].
Here,
we
recall
that
(b-1),
(b-2),
and
(b-3)
are
applied
in
[Lang1],
Chapter
IX,
§5,
to
verify
the
Universal
Norm
Index
Equality
for
cyclic
extensions.
This
Universal
Norm
Index
Equality
is
then
applied
in
[Lang1],
Chapter
X,
§1,
and
combined
with
the
theory
of
cyclotomic
extensions
in
[Lang1],
Chapter
X,
§2,
to
verify
(b-4).
Finally,
(b-4)
is
combined
with
(b-5)
in
[Lang1],
Chapter
XI,
§2,
to
complete
the
proof
of
the
Existence
Theorem
for
class
fields.
(ii)
From
the
point
of
view
of
the
theory
of
the
present
series
of
papers,
(a),
together
with
(b-3),
is
reminiscent
of
the
elementary
algebraic
number
theory
char-
acterization
of
nonzero
global
integers
as
roots
of
unity,
which
plays
an
im-
portant
role
in
the
theory
of
the
present
series
of
papers
[cf.
[IUTchIII],
the
proof
of
Proposition
3.10].
Moreover,
(a)
is
also
reminiscent
of
the
arithmetic
degrees
of
line
bundles
that
appear,
for
instance,
in
the
form
of
global
realified
Frobe-
nioids,
throughout
the
theory
of
the
present
series
of
papers.
Next,
we
observe
that
(b-1)
is
reminiscent
of
the
p-adic
absolute
anabelian
geometry
of
[AbsTopIII]
[cf.,
e.g.,
[AbsTopIII],
Corollary
1.10,
(i)].
On
the
other
hand,
(b-2)
is
reminiscent
of
repeated
applications
of
the
Prime
Number
Theorem
in
the
present
paper
[cf.
Propositions
1.6;
2.1,
(ii)];
this
comparison
between
(b-2)
and
the
Prime
Num-
ber
Theorem
will
be
discussed
in
more
detail
in
(iv)
below.
Next,
we
observe
[cf.
the
discussion
of
the
latter
portion
of
[IUTchIII],
Remark
3.12.1,
(iii)]
that
(b-4)
is
×
Z
=
{1}”
in
the
reminiscent
of
the
application
of
the
elementary
fact
“Q
>0
multiradial
algorithms
for
cyclotomic
rigidity
isomorphisms
in
the
number
field
case
[cf.
[IUTchI],
Example
5.1,
(v),
as
well
as
the
discussion
of
[IUTchIII],
Remarks
2.3.2,
2.3.3],
that
is
to
say,
not
only
in
the
sense
that
both
are
closely
related
to
the
various
cyclotomes
that
appear
in
global
class
field
theory
or
inter-universal
Teichmüller
theory,
but
also
in
the
sense
that
both
may
be
regarded
as
analogues
of
the
usual
product
formula
[i.e.,
which
appears
at
the
level
of
Frobenius-like
monoids
isomorphic
to
the
multiplicative
group
of
nonzero
elements
of
a
number
field]
at
the
level
of
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
59
certain
[étale-like!]
profinite
Galois
groups
related
to
global
number
fields.
On
the
other
hand,
(b-5)
is
reminiscent
of
the
central
role
played
through
inter-
universal
Teichmüller
theory
by
constructions
modeled
on
classical
Kummer
the-
ory.
In
fact,
these
comparisons
involving
(b-4)
and
(b-5)
are
closely
related
to
one
another
and
will
be
discussed
in
more
detail
in
(v),
(vi),
and
(vii)
below.
Next,
we
recall
that
Hodge-Tate
decompositions
as
in
(c)
play
a
central
role
in
the
proofs
of
the
main
results
of
[pGC],
which,
in
turn,
underlie
the
theory
of
[Ab-
sTopIII].
The
ramification
computations
concerning
finite
flat
group
schemes
as
in
(d)
are
reminiscent
of
various
p-adic
ramification
computations
concerning
log-
shells
in
[AbsTopIII],
as
well
as
in
Propositions
1.1,
1.2,
1.3,
1.4
of
the
present
paper.
Whereas
[Falt]
revolves
around
the
abelian/linear
theory
of
abelian
varieties
[cf.
(e)],
the
theory
of
the
present
series
of
papers
depends,
in
an
essential
way,
on
various
intricate
manipulations
involving
finite
étale
coverings
of
hyperbolic
curves,
such
as
the
use
of
Belyi
maps
in
[GenEll],
as
well
as
in
the
Belyi
cuspidal-
izations
applied
in
[AbsTopIII].
The
theory
of
polarizations
of
abelian
varieties
applied
in
[Falt]
[cf.
(f)]
is
reminiscent
of
the
essential
role
played
by
commutators
of
theta
groups
in
the
theory
of
[EtTh],
which,
in
turn,
plays
a
central
role
in
the
theory
of
the
present
series
of
papers.
Finally,
the
logarithmic
geometry
of
(g)
is
reminiscent
of
the
combinatorial
anabelian
geometry
of
[SemiAnbd],
which
is
applied,
in
[IUTchI],
§2,
to
the
logarithmic
geometry
of
coverings
of
stable
curves.
(iii)
One
way
to
summarize
the
discussion
of
(ii)
is
as
follows:
many
aspects
of
the
theory
of
[Falt]
may
be
regarded
as
“distant
abelian
ancestors”
of
certain
aspects
of
the
“anabelian-based
theory”
of
inter-
universal
Teichmüller
theory.
Alternatively,
one
may
observe
that
the
overwhelmingly
scheme-theoretic
nature
of
the
theory
applied
in
[Falt]
lies
in
stark
contrast
to
the
highly
non-scheme-theoretic
nature
of
the
absolute
anabelian
geometry
and
theory
of
monoids/Frobenioids
ap-
plied
in
the
present
series
of
papers:
that
is
to
say,
many
aspects
of
the
theory
of
[Falt]
may
be
regarded
as
“distant
arith-
metically
holomorphic
ancestors”
of
certain
aspects
of
the
multira-
dial
and
mono-analytic
[i.e.,
“arithmetically
real
analytic”]
theory
developed
in
inter-universal
Teichmüller
theory.
One
way
to
understand
this
fundamental
difference
between
the
theory
of
[Falt]
and
inter-universal
Teichmüller
theory
is
by
considering
the
naive
goal
of
construct-
ing
some
sort
of
“Frobenius
morphism”
on
a
number
field
[cf.
the
discussion
of
[FrdI],
§I3],
i.e.,
which
has
the
effect
of
multiplying
arithmetic
degrees
by
a
positive
factor
>
1:
whereas
the
theory
of
[Falt]
[cf.,
e.g.,
the
argument
of
the
proof
of
[GenEll],
Lemma
3.5,
as
discussed
in
Remark
2.3.2,
(ii)]
may
regarded
as
a
reflection
of
the
point
of
view
that,
so
long
as
one
respects
the
arithmetic
holomorphic
structure
of
scheme
theory,
such
a
“Frobenius
morphism”
on
a
number
field
cannot
exist,
the
essential
content
of
inter-universal
Teichmüller
theory
may
be
summarized
in
a
word
as
the
assertion
that,
60
SHINICHI
MOCHIZUKI
if
one
dismantles
this
arithmetic
holomorphic
structure
in
a
suitably
canon-
ical
fashion
and
allows
oneself
to
work
with
multiradial/mono-analytic
[i.e.,
“arithmetically
quasi-conformal”]
structures,
then
one
can
in-
deed
construct,
in
a
very
canonical
fashion,
such
a
“Frobenius
mor-
phism”
on
a
number
field.
(iv)
In
the
context
of
the
comparison
discussed
in
(ii)
concerning
(b-2),
it
is
of
interest
to
note
that
the
fundamental
difference
discussed
in
(iii)
between
the
theory
of
[Falt]
and
inter-universal
Teichmüller
theory
is,
in
some
sense,
reflected
in
the
difference
between
the
theory
of
global
density
of
primes
[i.e.,
(b-2)]
and
the
Prime
Number
Theorem.
That
is
to
say,
the
coherence
of
the
sorts
of
collections
of
primes
that
appear
in
the
theory
of
global
density
of
primes
may
be
thought
of
as
a
sort
of
representation,
in
the
context
of
analytic
number
theory,
of
the
arithmetic
holomorphic
structure
of
conventional
scheme
theory.
By
contrast,
in
the
context
of
the
Prime
Number
Theorem,
primes
of
a
number
field
appear,
so
to
speak,
one
by
one,
i.e.,
in
a
fashion
that
is
only
possible
if
one
deactivates,
in
the
context
of
analytic
number
theory,
the
coherence
that
underlies
the
aggregrations
of
primes
that
appear
in
the
theory
of
global
density
of
primes.
That
is
to
say,
this
approach
to
treating
primes
“one
by
one”
may
be
thought
of
as
corresponding
to
the
dismantling
of
arithmetic
holomorphic
structures
that
occurs
in
the
context
of
the
multiradial/mono-analytic
structures
that
appear
in
inter-universal
Teichmüller
theory.
Here,
it
is
also
of
interest
to
note
that
the
way
in
which
one
“deactivates
aggregations
of
primes”
in
the
context
of
the
Prime
Number
Theorem
may
be
thought
of
[cf.
the
discussion
of
[IUTchIII],
Remark
3.12.2,
(i),
(c)]
as
a
sort
of
dismantling
of
the
ring
structure
of
a
number
field
into
its
underlying
additive
[i.e.,
counting
primes
“one
by
one”!]
and
multiplicative
structures
[i.e.,
the
very
notion
of
a
prime!].
(v)
The
fundamental
difference
discussed
in
(iii)
between
the
theory
of
[Falt]
and
inter-universal
Teichmüller
theory
may
also
be
seen
in
the
context
of
the
com-
parison
discussed
in
(ii)
concerning
(b-4).
Indeed,
the
global
reciprocity
law
of
(b-4),
which
plays
a
central
role
in
global
class
field
theory,
depends,
in
an
essential
way,
on
nontrivial
relationships
between
local
units
[such
as
the
unit
determined
by
a
prime
number
l
at
a
nonarchimedean
prime
of
a
number
field
of
residue
char-
acteristic
=
l]
at
one
prime
of
a
number
field
and
elements
of
local
value
groups
[such
as
the
element
determined
by
l
at
a
nonarchimedean
prime
of
a
number
field
of
residue
characteristic
l]
at
another
prime
of
the
number
field.
Such
nontrivial
relationships
are
fundamentally
incompatible
with
the
splittings/decouplings
of
lo-
cal
units
and
local
value
groups
that
play
a
central
role
in
the
dismantling
of
arithmetic
holomorphic
structures
that
occurs
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
[IUTchIII],
Remark
2.3.3,
(i);
[IUTchIII],
Remark
3.12.2,
(i),
(a)].
This
incompatibility
[i.e.,
with
nontrivial
relationships
between
local
units
and
local
value
groups
at
nonarchimedean
primes
with
distinct
residue
characteristics]
may
also
be
seen
quite
explicitly
in
the
structure
of
the
various
types
of
prime-strips
that
appear
in
inter-universal
Teichmüller
theory
[cf.
[IUTchI],
Fig.
I1.2].
That
is
to
say,
such
nontrivial
relationships,
which
form
the
content
of
the
global
reciprocity
law
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
61
of
global
class
field
theory,
may
be
thought
of
as
a
sort
of
global
Galois-theoretic
representation
of
the
constraints
that
constitute
the
arithmetic
holomorphic
structure
of
conventional
scheme
theory.
(vi)
Another
fundamental
aspect
of
the
comparison
discussed
in
(ii)
concern-
ing
(b-4)
may
be
seen
in
the
fact
that
whereas
the
global
reciprocity
law
of
global
class
field
theory
concerns
the
global
reciprocity
map,
the
cyclotomic
rigid-
ity
algorithms
of
inter-universal
Teichmüller
theory
to
which
(b-4)
was
compared
appear
in
the
context
of
Kummer-theoretic
isomorphisms.
That
is
to
say,
although
both
the
global
reciprocity
map
and
Kummer-theoretic
isomorphisms
in-
volve
correspondences
between
multiplicative
monoids
associated
to
number
fields
and
multiplicative
monoids
that
arise
from
global
Galois
groups,
one
fundamental
difference
between
these
two
types
of
correspondence
lies
in
the
fact
that
whereas
Kummer-theoretic
isomorphisms
satisfy
very
strong
covariant
[with
respect
to
functions]
functoriality
properties,
the
reciprocity
maps
that
appear
in
various
versions
of
class
field
theory
tend
not
to
satisfy
such
strong
functoriality
properties.
This
presence
or
absence
of
strong
functoriality
properties
is,
to
a
substantial
extent,
a
reflection
of
the
fact
that
whereas
Kummer
theory
may
be
performed
in
a
very
straightforward,
tautological,
“general
nonsense”
fashion
in
a
wide
variety
of
situations,
class
field
theory
may
only
be
conducted
in
very
special
arithmetic
situations.
This
presence
of
strong
functoriality
properties
[i.e.,
in
the
case
of
Kummer
theory]
is
the
essential
reason
for
the
central
role
played
by
Kummer
theory
[cf.
(b-5)]
in
inter-universal
Teichmüller
theory,
as
well
as
in
many
situations
that
arise
in
anabelian
geometry
in
general
[cf.,
e.g.,
the
theory
of
[Cusp]].
Indeed,
the
very
tautological/ubiquitous/strongly
functorial
nature
of
Kummer
theory
makes
it
well-suited
to
the
sort
of
dismantling
of
ring
structures
that
occurs
in
inter-universal
Teichmüller
theory,
as
well
as
to
the
various
evaluation
operations
of
functions
at
special
points
that
play
a
central
role,
in
the
context
of
Galois
evaluation,
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
[IUTchIII],
Remark
2.3.3].
By
contrast,
although
there
exist
various
higher-dimensional
ver-
sions
of
class
field
theory
involving
higher
algebraic
K-groups,
these
versions
of
class
field
theory
are
fundamentally
incompatible
with
the
crucial
evaluation
of
function
operations
of
the
sort
that
occur
in
inter-universal
Teichmüller
theory.
Indeed,
more
generally,
except
for
very
exceptional
classical
cases
involving
exponential
functions
in
the
case
of
Q
or
modular
and
elliptic
functions
in
the
case
of
imaginary
quadratic
fields,
class
field
theory
tends
to
be
very
ill-suited
to
situations
that
involve
the
evaluation
of
special
functions
at
special
points.
Moreover,
even
if
one
restricts
one’s
attention,
for
instance,
to
functoriality
with
respect
to
passing
to
a
finite
extension
field,
the
functoriality
of
the
reciprocity
maps
that
occur
in
class
field
theory
are
[unlike
Kummer-theoretic
isomorphisms!]
contravariant
[with
respect
to
functions]
and
can
only
be
made
covariant
if
one
applies
some
sort
of
nontrivial
duality
result
to
reverse
the
direction
of
the
maps
—
62
SHINICHI
MOCHIZUKI
a
state
of
affairs
that
makes
class
field
theory
very
difficult
to
apply
not
only
in
inter-
universal
Teichmüller
theory,
but
also
in
many
situations
that
arise
in
anabelian
geometry.
On
the
other
hand,
in
the
context
of
inter-universal
Teichmüller
theory,
the
price,
so
to
speak,
that
one
pays
for
the
very
convenient,
“general
nonsense”
nature
of
Kummer
theory
lies
in
the
highly
nontrivial
nature
—
which
may
be
seen,
for
instance,
in
the
establishment
of
various
multiradiality
properties
—
of
the
cyclotomic
rigidity
algorithms
that
appear
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
[IUTchIII],
Remark
2.3.3].
Here,
we
recall
that
such
cyclotomic
rigidity
algorithms
—
which
never
appear
in
discussions
of
conventional
arithmetic
geometry
in
which
the
arithmetic
holo-
morphic
structure
is
held
fixed
—
play
a
central
role
in
inter-universal
Teichmüller
theory
precisely
because
of
the
indeterminacies
that
arise
as
a
consequence
of
the
dismantling
of
the
arithmetic
holomorphic
structure.
Finally,
in
this
context,
it
is
of
interest
to
recall
that,
although
local
class
field
theory
is,
in
a
certain
limited
sense,
applied
in
inter-universal
Teichmüller
theory,
i.e.,
in
order
to
obtain
cyclotomic
rigidity
algorithms
for
MLF-Galois
pairs
[cf.
[IUTchII],
Proposition
1.3,
(ii)],
it
is
only
“of
limited
use”
in
the
sense
that
the
resulting
cyclotomic
rigidity
algorithms
are
uniradial
[i.e.,
fail
to
be
multiradial
—
cf.
[IUTchIII],
Figs.
2.1,
3.7,
and
the
surrounding
discussions].
(vii)
The
fundamental
incompatibility
—
i.e.,
except
in
very
exceptional
clas-
sical
cases
involving
exponential
functions
in
the
case
of
Q
or
modular
and
elliptic
functions
in
the
case
of
imaginary
quadratic
fields
—
discussed
in
(vi)
of
class
field
theory
with
situations
that
involve
the
evaluation
of
special
functions
at
special
points
is
highly
reminiscent
of
the
original
point
of
view
of
class
field
theory
in
the
early
twentieth
century
[cf.
Kronecker’s
Jugendtraum,
Hilbert’s
twelfth
problem],
i.e.,
to
the
effect
that
further
development
of
class
field
theory
should
proceed
precisely
by
extending
the
theory
involving
evaluation
of
special
functions
at
special
points
that
exists
in
these
“exceptional
classical
cases”
to
the
case
of
ar-
bitrary
number
fields.
This
state
of
affairs
is,
in
turn,
highly
reminiscent
of
the
fact
that
the
approach
taken
in
the
above
discussion
to
“dissecting
global
class
field
the-
ory”
is
the
oldest/original
approach
to
global
class
field
theory,
as
well
as
of
the
fact
that
this
original
approach
is
the
most
well-suited
to
discussions
of
comparisons
between
the
theory
of
[Falt]
and
inter-universal
Teichmüller
theory.
This
state
of
affairs
is
also
highly
reminiscent
of
the
discussion
in
[Pano],
§3,
§4,
of
the
numerous
analogies
between
inter-universal
Teichmüller
theory
and
the
classical
[i.e.,
dating
back
to
the
nineteenth
century!]
theory
surrounding
Jacobi’s
identity
for
the
theta
function
on
the
upper
half-plane
and
Gaussian
distributions/integrals.
Finally,
this
collection
of
observations,
taken
as
a
whole,
may
be
summarized
as
follows:
Many
of
the
ideas
that
appear
in
inter-universal
Teichmüller
theory
bear
a
much
closer
resemblance
to
the
mathematics
of
the
late
nine-
teenth
and
early
twentieth
centuries
—
i.e.,
to
the
mathematics
of
Gauss,
Jacobi,
Kummer,
Kronecker,
Weber,
Frobenius,
Hilbert,
and
Teichmüller
—
than
to
the
mathematics
of
the
mid-
to
late
twen-
tieth
century.
This
close
resemblance
suggests
strongly
that,
relative
to
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
63
the
mathematics
of
the
late
nineteenth
and
early
twentieth
centuries,
the
course
of
development
of
a
substantial
portion
of
the
mathematics
of
the
mid-
to
late
twentieth
century
should
not
be
regarded
as
“unique”
or
“inevitable”,
but
rather
as
being
merely
one
possible
choice
among
many
viable
and
fruitful
alternatives
that
existed
a
priori.
Here,
we
note
that
although
the
use,
in
inter-universal
Teichmüller
theory,
of
Belyi
maps,
as
well
as
of
the
p-adic
anabelian
geometry
of
the
1990’s
[i.e.,
[pGC]],
may
at
first
glance
look
like
an
incidence
of
“exceptions”
to
the
“rule”
constituted
by
this
point
of
view,
these
“exceptions”
may
be
thought
of
as
“proving
the
rule”
in
the
sense
that
they
are
far
from
typical
of
the
mathematics
of
the
late
twentieth
century.
Remark
2.3.4.
Various
aspects
of
the
theory
of
the
present
series
of
papers
are
substantially
reminiscent
of
the
theory
surrounding
Bogomolov’s
proof
of
the
geometric
version
of
the
Szpiro
Conjecture,
as
discussed
in
[ABKP],
[Zh].
Put
another
way,
these
aspects
of
the
theory
of
the
present
series
of
papers
may
be
thought
of
as
arithmetic
analogues
of
the
geometric
theory
surrounding
Bo-
gomolov’s
proof.
Alternatively,
Bogomolov’s
proof
may
be
thought
of
as
a
sort
of
useful
elementary
guide,
or
blueprint
[perhaps
even
a
sort
of
Rosetta
stone!],
for
understanding
substantial
portions
of
the
theory
of
the
present
series
of
papers.
The
author
would
like
to
express
his
gratitude
to
Ivan
Fesenko
for
bringing
to
his
attention,
via
numerous
discussions
in
person,
e-mails,
and
skype
conversations
between
December
2014
and
January
2015,
the
possibility
of
the
existence
of
such
fascinating
connections
between
Bogomolov’s
proof
and
the
theory
of
the
present
series
of
papers.
We
discuss
these
analogies
in
more
detail
in
[BogIUT].
Remark
2.3.5.
In
[Par],
a
proof
is
given
of
the
Mordell
Conjecture
for
function
fields
over
the
complex
numbers.
Like
the
proof
of
Bogomolov
discussed
in
Remark
2.3.4,
Parshin’s
proof
involves
metric
estimates
of
“displacements”
that
arise
from
actions
of
elements
of
the
[usual
topological]
fundamental
group
of
the
complex
hyperbolic
curve
that
serves
as
the
base
scheme
of
the
given
family
of
curves.
In
particular,
we
observe
that
one
may
pose
the
following
question:
Is
it
possible
to
apply
some
portion
of
the
ideas
of
the
inter-universal
Teichmüller
theory
developed
in
the
present
series
of
papers
to
obtain
a
proof
of
the
Mordell
Conjecture
over
number
fields
without
making
use
of
Belyi
maps
as
in
the
proof
of
Corollary
2.3
[i.e.,
the
proof
of
[GenEll],
Theorem
2.1]?
This
question
was
posed
to
the
author
by
Felipe
Voloch
in
an
e-mail
in
September
2015.
The
answer
to
this
question
is,
as
far
as
the
author
can
see
at
the
time
of
writing,
“no”.
On
the
other
hand,
this
question
is
interesting
in
the
context
of
the
discussion
of
Remarks
2.3.3
and
2.3.4
in
that
it
serves
to
highlight
various
inter-
esting
aspects
of
inter-universal
Teichmüller
theory,
as
we
explain
in
the
following
discussion.
(i)
First,
we
recall
[cf.,
e.g.,
[Lang2],
Chapter
I,
§1,
§2,
for
more
details]
that
the
starting
point
of
the
theory
of
the
Kobayashi
distance
on
a
[Kobayashi]
hyperbolic
complex
manifold
is
the
well-known
Schwarz
lemma
of
elementary
complex
analysis
64
SHINICHI
MOCHIZUKI
and
its
consequences
for
the
geometry
of
holomorphic
maps
from
the
open
unit
disc
D
in
the
complex
plane
to
an
arbitrary
complex
manifold.
In
the
following
discussion,
we
shall
refer
to
this
geometry
as
the
Schwarz-theoretic
geometry
of
D.
Perhaps
the
most
fundamental
difference
between
the
proofs
of
Parshin
and
Bogomolov
lies
in
the
fact
that
(PB1)
Whereas
Parshin’s
proof
revolves
around
estimates
of
displacements
aris-
ing
from
actions
of
elements
of
the
fundamental
group
on
a
certain
two-
dimensional
complete
[Kobayashi]
hyperbolic
complex
manifold
by
means
of
the
holomorphic
geometry
of
the
Kobayashi
distance,
i.e.,
in
effect,
the
Schwarz-theoretic
geometry
of
D,
Bogomolov’s
proof
[cf.
the
re-
view
of
Bogomolov’s
proof
given
in
[BogIUT]]
revolves
around
estimates
of
displacements
arising
from
actions
of
elements
of
the
fundamental
group
on
a
one-dimensional
real
analytic
manifold
[i.e.,
a
universal
covering
of
a
copy
of
the
unit
circle
S
1
]
by
means
of
the
real
analytic
symplectic
geometry
of
the
upper
half-plane.
Here,
it
is
already
interesting
to
note
that
this
fundamental
gap,
in
the
case
of
results
over
complex
function
fields,
between
the
holomorphic
geometry
applied
in
Parshin’s
proof
of
the
Mordell
Conjecture
and
the
real
analytic
symplectic
geometry
applied
in
Bogomolov’s
proof
of
the
Szpiro
Conjecture
is
highly
reminiscent
of
the
fundamental
gap
discussed
in
Remark
2.3.3,
(iii),
in
the
case
of
results
over
number
fields,
between
the
arithmetically
holomorphic
nature
of
the
proof
of
the
Mordell
Conjecture
given
in
[Falt]
and
the
“arithmetically
quasi-conformal”
nature
of
the
proof
of
the
Szpiro
Conjecture
[cf.
Corollary
2.3]
via
inter-universal
Teichmüller
theory
given
in
the
present
series
of
papers.
That
is
to
say,
Parshin’s
proof
is
best
understood
not
as
a
“weaker,
or
simplified,
ver-
sion
of
Bogomolov’s
proof
obtained
by
extracting
certain
portions
of
Bogo-
molov’s
proof
”,
but
rather
as
a
proof
that
reflects
a
fundamentally
quali-
tatively
different
geometry
—
i.e.,
holomorphic,
as
opposed
to
real
an-
alytic
—
from
Bogomolov’s
proof.
This
point
of
view
already
suggests
rather
strongly,
relative
to
the
analogy
between
Bogomolov’s
proof
and
inter-universal
Teichmüller
theory
[cf.
[BogIUT]]
that
it
is
unnatural/unrealistic
to
expect
to
obtain
a
new
proof
of
the
Mordell
Conjecture
over
number
fields
by
applying
some
portion
of
the
ideas
of
the
inter-universal
Teichmüller
theory.
(ii)
At
a
more
technical
level,
the
fundamental
difference
(PB1)
discussed
in
(i)
may
be
seen
in
the
fact
that
(PB2)
whereas
Parshin’s
proof
involves
numerous
holomorphic
maps
from
the
open
unit
disc
D
into
one-
and
two-dimensional
complex
manifolds
[i.e.,
in
essence,
the
universal
coverings
of
the
base
space
and
total
space
of
the
family
of
curves
under
consideration],
Bogomolov’s
proof
revolves
around
the
real
analytic
symplectic
geometry
of
a
fixed
copy
of
the
open
unit
disc
D
[or,
equivalently,
the
upper
half-plane],
i.e.,
in
Bogomolov’s
proof,
one
never
considers
holomorphic
maps
from
D
to
itself
which
are
not
biholomorphic.
The
essentially
arbitrary
nature
of
these
numerous
holomorphic
maps
that
appear
in
Parshin’s
proof
is
reflected
in
the
fact
that
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
(PB3)
65
Parshin’s
proof
is
well-suited
to
proving
a
rough
qualitative
[i.e.,
“finiteness”]
result
for
families
of
curves
of
arbitrary
genus
≥
2,
whereas
Bogomolov’s
proof
is
well-suited
to
proving
a
much
finer
explicit
inequal-
ity,
but
only
in
the
case
of
families
of
elliptic
curves.
Another
technical
aspect
of
the
proofs
of
Parshin
and
Bogomolov
that
is
closely
related
to
both
(PB2)
and
(PB3)
is
the
fact
that
(PB4)
whereas
the
estimation
apparatus
of
Bogomolov’s
proof
depends
in
an
essential
way
on
special
properties
of
particular
types
of
elements
—
such
as
unipotent
elements
or
commutators
—
of
the
fundamental
group
under
consideration,
the
estimation
apparatus
of
Parshin’s
proof
is
uni-
form
for
arbitrary
[“sufficiently
small”]
elements
of
the
fundamental
group
under
consideration.
(iii)
Although,
as
discussed
in
(ii),
it
is
difficult
to
see
how
Parshin’s
proof
could
be
“embedded”
into
[i.e.,
obtained
as
a
“suitable
portion
of
”]
Bogomolov’s
proof,
the
Schwarz-theoretic
geometry
of
D
admits
a
“natural
embedding”
into
[i.e.,
admits
a
natural
analogy
to
a
suitable
portion
of]
inter-universal
Teichmüller
theory,
namely,
in
the
form
of
the
theory
of
categories
of
localizations
of
the
sort
that
appear
in
[GeoAnbd],
§2;
[AbsTopI],
§4;
[AbsTopII],
§3.
This
theory
of
cate-
gories
of
localizations
culminates
in
the
theory
of
Belyi
cuspidalizations,
which
is
discussed
in
[AbsTopII],
§3,
and
applied
to
obtained
the
mono-anabelian
recon-
struction
algorithms
of
[AbsTopIII],
§1.
Moreover,
the
analogy
between
such
categories
of
localizations
and
the
classical
Schwarz-theoretic
geometry
of
D
[or,
equivalently,
the
upper
half-plane]
is
discussed
in
the
Introduction
to
[GeoAnbd],
as
well
as
in
[IUTchI],
Remark
5.1.4.
This
theory
of
categories
of
localizations
may
be
summarized
roughly
as
follows:
In
the
context
of
absolute
anabelian
geometry
over
number
fields
and
their
nonarchimedean
localizations,
Belyi
maps
play
the
role
of
the
Schwarz-theoretic
geometry
of
the
open
unit
disc
D,
i.e.,
the
role
of
realizing
a
sort
of
arithmetic
version
of
analytic
continuation.
This
point
of
view
is
also
interesting
from
the
point
of
view
of
the
discussion
of
Remark
2.2.4,
(iii),
i.e.,
to
the
effect
that
[noncritical]
Belyi
maps
play
the
role
of
realizing
a
sort
of
arithmetic
version
of
analytic
continuation
in
the
proof
of
[GenEll],
Theorem
2.1.
That
is
to
say,
from
the
point
of
view
of
the
question
posed
at
the
beginning
of
the
present
Remark
2.3.5:
Even
if,
in
the
context
of
inter-universal
Teichmüller
theory,
one
attempts
to
search
for
an
analogue
of
Parshin’s
proof
in
the
form
of
a
“suitable
portion”
of
the
inter-universal
Teichmüller
theory
developed
in
[IUTchI],
[IUTchII],
[IUTchIII]
[i.e.,
even
if
one
avoids
consideration
of
the
applica-
tion
of
[noncritical]
Belyi
maps
in
the
proof
of
Corollary
2.3
via
[GenEll],
Theorem
2.1],
one
is
ultimately
led
—
i.e.,
from
the
point
of
view
of
con-
sidering
arithmetic
analogues
of
the
classical
complex
theory
of
analytic
continuation
and
the
Schwarz-theoretic
geometry
of
the
open
unit
disc
D
—
to
the
Belyi
maps
that
appear
in
the
Belyi
cuspidalizations
of
[AbsTopII],
§3;
[AbsTopIII],
§1.
66
SHINICHI
MOCHIZUKI
Put
another
way,
it
appears
that
any
search
in
the
realm
of
inter-universal
Te-
ichmüller
theory
either
for
some
proof
of
the
Mordell
Conjecture
[over
number
fields]
or
for
some
analogue
of
Parshin’s
proof
[of
the
Mordell
Conjecture
over
com-
plex
function
fields]
appears
to
lead
inevitably
to
some
application
of
Belyi
maps
to
realize
some
sort
of
arithmetic
analogue
of
the
classical
complex
theory
of
ana-
lytic
continuation
and
the
Schwarz-theoretic
geometry
of
the
open
unit
disc
D.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
67
Section
3:
Inter-universal
Formalism:
the
Language
of
Species
In
the
present
§3,
we
develop
—
albeit
from
an
extremely
naive/non-expert
point
of
view,
relative
to
the
theory
of
foundations!
—
the
language
of
species.
Roughly
speaking,
a
“species”
is
a
“type
of
mathematical
object”,
such
as
a
“group”,
a
“ring”,
a
“scheme”,
etc.
In
some
sense,
this
language
may
be
thought
of
as
an
explicit
description
of
certain
tasks
typically
executed
at
an
implicit,
intuitive
level
by
mathematicians
[i.e.,
mathematicians
who
are
not
equipped
with
a
detailed
knowledge
of
the
theory
of
foundations!]
via
a
sort
of
“mental
arithmetic”
in
the
course
of
interpreting
various
mathematical
arguments.
In
the
context
of
the
theory
developed
in
the
present
series
of
papers,
however,
it
is
useful
to
describe
these
intuitive
operations
explicitly.
In
the
following
discussion,
we
shall
work
with
various
models
—
consisting
of
“sets”
and
a
relation
“∈”
—
of
the
standard
ZFC
axioms
of
axiomatic
set
theory
[i.e.,
the
nine
axioms
of
Zermelo-Fraenkel,
together
with
the
axiom
of
choice
—
cf.,
e.g.,
[Drk],
Chapter
1,
§3].
We
shall
refer
to
such
models
as
ZFC-models.
Recall
that
a
(Grothendieck)
universe
V
is
a
set
satisfying
the
following
axioms
[cf.
[McLn],
p.
194]:
(i)
V
is
transitive,
i.e.,
if
y
∈
x,
x
∈
V
,
then
y
∈
V
.
(ii)
The
set
of
natural
numbers
N
∈
V
.
(iii)
If
x
∈
V
,
then
the
power
set
of
x
also
belongs
to
V
.
(iv)
If
x
∈
V
,
then
the
union
of
all
members
of
x
also
belongs
to
V
.
(v)
If
x
∈
V
,
y
⊆
V
,
and
f
:
x
→
y
is
a
surjection,
then
y
∈
V
.
We
shall
say
that
a
set
E
is
a
V
-set
if
E
∈
V
.
The
various
ZFC-models
that
we
work
with
may
be
thought
of
as
[but
are
not
restricted
to
be!]
the
ZFC-models
determined
by
various
universes
that
are
sets
relative
to
some
ambient
ZFC-model
which,
in
addition
to
the
standard
ax-
ioms
of
ZFC
set
theory,
satisfies
the
following
existence
axiom
[attributed
to
the
“Grothendieck
school”
—
cf.
the
discussion
of
[McLn],
p.
193]:
(†
G
)
Given
any
set
x,
there
exists
a
universe
V
such
that
x
∈
V
.
We
shall
refer
to
a
ZFC-model
that
also
satisfies
this
additional
axiom
of
the
Grothendieck
school
as
a
ZFCG-model.
This
existence
axiom
(†
G
)
implies,
in
par-
ticular,
that:
Given
a
set
I
and
a
collection
of
universes
V
i
,
where
i
∈
I,
indexed
by
I
[i.e.,
a
‘function’
I
i
→
V
i
],
there
exists
a
[larger]
universe
V
such
that
V
i
∈
V
,
for
i
∈
I.
Indeed,
since
the
graph
of
the
function
I
i
→
V
i
is
a
set,
it
follows
that
{V
i
}
i∈I
is
a
set.
Thus,
it
follows
from
the
existence
axiom
(†
G
)
that
there
exists
a
universe
V
such
that
{V
i
}
i∈I
∈
V
.
Hence,
by
condition
(i),
we
conclude
that
V
i
∈
V
,
for
all
i
∈
I,
as
desired.
Note
that
this
means,
in
particular,
that
there
exist
infinite
ascending
chains
of
universes
V
0
∈
V
1
∈
V
2
∈
V
3
∈
.
.
.
∈
V
n
∈
.
.
.
∈
V
68
SHINICHI
MOCHIZUKI
—
where
n
ranges
over
the
natural
numbers.
On
the
other
hand,
by
the
axiom
of
foundation,
there
do
not
exist
infinite
descending
chains
of
universes
V
0
V
1
V
2
V
3
.
.
.
V
n
.
.
.
—
where
n
ranges
over
the
natural
numbers.
Although
we
shall
not
discuss
in
detail
here
the
quite
difficult
issue
of
whether
or
not
there
actually
exist
ZFCG-models,
we
remark
in
passing
that
it
may
be
possible
to
justify
the
stance
of
ignoring
such
issues
in
the
context
of
the
present
series
of
papers
—
at
least
from
the
point
of
view
of
establishing
the
validity
of
various
“final
results”
that
may
be
formulated
in
ZFC-models
—
by
invoking
the
work
of
Feferman
[cf.
[Ffmn]].
Precise
statements
concerning
such
issues,
however,
lie
beyond
the
scope
of
the
present
paper
[as
well
as
of
the
level
of
expertise
of
the
author!].
In
the
following
discussion,
we
use
the
phrase
“set-theoretic
formula”
as
it
is
conventionally
used
in
discussions
of
axiomatic
set
theory
[cf.,
e.g.,
[Drk],
Chapter
1,
§2],
with
the
following
proviso:
In
the
following
discussion,
it
should
be
understood
that
every
set-theoretic
formula
that
appears
is
“absolute”
in
the
sense
that
its
validity
for
a
collection
of
sets
contained
in
some
universe
V
relative
to
the
model
of
set
theory
determined
by
V
is
equivalent,
for
any
universe
W
such
that
V
∈
W
,
to
its
validity
for
the
same
collection
of
sets
relative
to
the
model
of
set
theory
determined
by
W
[cf.,
e.g.,
[Drk],
Chapter
3,
Definition
4.2].
Definition
3.1.
(i)
A
0-species
S
0
is
a
collection
of
conditions
given
by
a
set-theoretic
formula
Φ
0
(E)
involving
an
ordered
collection
E
=
(E
1
,
.
.
.
,
E
n
0
)
of
sets
E
1
,
.
.
.
,
E
n
0
[which
we
think
of
as
“indeterminates”],
for
some
integer
n
0
≥
1;
in
this
situation,
we
shall
refer
to
E
as
a
collection
of
species-data
for
S
0
.
If
S
0
is
a
0-species
given
by
a
set-theoretic
formula
Φ
0
(E),
then
a
0-specimen
of
S
0
is
a
specific
ordered
collection
of
n
0
sets
E
=
(E
1
,
.
.
.
,
E
n
0
)
in
some
specific
ZFC-model
that
satisfies
Φ
0
(E).
If
E
is
a
0-specimen
of
a
0-species
S
0
,
then
we
shall
write
E
∈
S
0
.
If,
moreover,
it
holds,
in
any
ZFC-model,
that
the
0-specimens
of
S
0
form
a
set,
then
we
shall
refer
to
S
0
as
0-small.
(ii)
Let
S
0
be
a
0-species.
Then
a
1-species
S
1
acting
on
S
0
is
a
collection
of
set-theoretic
formulas
Φ
1
,
Φ
1◦1
satisfying
the
following
conditions:
(a)
Φ
1
is
a
set-theoretic
formula
Φ
1
(E,
E
,
F)
involving
two
collections
of
species-data
E,
E
for
S
0
[i.e.,
the
conditions
Φ
0
(E),
Φ
0
(E
)
hold]
and
an
ordered
collection
F
=
(F
1
,
.
.
.
,
F
n
1
)
of
[“in-
determinate”]
sets
F
1
,
.
.
.
,
F
n
1
,
for
some
integer
n
1
≥
1;
in
this
situation,
we
shall
refer
to
(E,
E
,
F)
as
a
collection
of
species-data
for
S
1
and
write
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
69
F
:
E
→
E
.
If,
in
some
ZFC-model,
E,
E
∈
S
0
,
and
F
is
a
specific
or-
dered
collection
of
n
1
sets
that
satisfies
the
condition
Φ
1
(E,
E
,
F
),
then
we
shall
refer
to
the
data
(E,
E
,
F
)
as
a
1-specimen
of
S
1
and
write
(E,
E
,
F
)
∈
S
1
;
alternatively,
we
shall
denote
a
1-specimen
(E,
E
,
F
)
via
the
notation
F
:
E
→
E
and
refer
to
E
(respectively,
E
)
as
the
domain
(respectively,
codomain)
of
F
:
E
→
E
.
(b)
Φ
1◦1
is
a
set-theoretic
formula
Φ
1◦1
(E,
E
,
E
,
F,
F
,
F
)
involving
three
collections
of
species-data
F
:
E
→
E
,
F
:
E
→
E
,
F
:
E
→
E
for
S
1
[i.e.,
the
conditions
Φ
0
(E);
Φ
0
(E
);
Φ
0
(E
);
Φ
1
(E,
E
,
F);
Φ
1
(E
,
E
,
F
);
Φ
1
(E,
E
,
F
)
hold];
in
this
situation,
we
shall
refer
to
F
as
a
composite
of
F
with
F
and
write
F
=
F
◦
F
[which
is,
a
priori,
an
abuse
of
notation,
since
there
may
exist
many
composites
of
F
with
F
—
cf.
(c)
below];
we
shall
use
similar
terminology
and
notation
for
1-specimens
in
specific
ZFC-models.
(c)
Given
a
pair
of
1-specimens
F
:
E
→
E
,
F
:
E
→
E
of
S
1
in
some
ZFC-model,
there
exists
a
unique
composite
F
:
E
→
E
of
F
with
F
in
the
given
ZFC-model.
(d)
Composition
of
1-specimens
F
:
E
→
E
,
F
:
E
→
E
,
F
:
E
→
E
of
S
1
in
a
ZFC-model
is
associative.
(e)
For
any
0-specimen
E
of
S
0
in
a
ZFC-model,
there
exists
a
[necessarily
unique]
1-specimen
F
:
E
→
E
of
S
1
[in
the
given
ZFC-model]
—
which
we
shall
refer
to
as
the
identity
1-specimen
id
E
of
E
—
such
that
for
any
1-specimens
F
:
E
→
E,
F
:
E
→
E
of
S
1
[in
the
given
ZFC-model]
we
have
F
◦
F
=
F
,
F
◦
F
=
F
.
If,
moreover,
it
holds,
in
any
ZFC-model,
that
for
any
two
0-specimens
E,
E
of
S
0
,
the
1-specimens
F
:
E
→
E
of
S
1
[i.e.,
the
1-specimens
of
S
1
with
domain
E
and
codomain
E
]
form
a
set,
then
we
shall
refer
to
S
1
as
1-small.
(iii)
A
species
S
is
defined
to
be
a
pair
consisting
of
a
0-species
S
0
and
a
1-
species
S
1
acting
on
S
0
.
Fix
a
species
S
=
(S
0
,
S
1
).
Let
i
∈
{0,
1}.
Then
we
shall
refer
to
an
i-specimen
of
S
i
as
an
i-specimen
of
S.
We
shall
refer
to
a
0-specimen
(respectively,
1-specimen)
of
S
as
a
species-object
(respectively,
a
species-morphism)
of
S.
We
shall
say
that
S
is
i-small
if
S
i
is
i-small.
We
shall
refer
to
a
species-
morphism
F
:
E
→
E
as
a
species-isomorphism
if
there
exists
a
species-morphism
F
:
E
→
E
such
that
the
composites
F
◦F
,
F
◦F
are
identity
species-morphisms;
in
this
situation,
we
shall
say
that
E,
E
are
species-isomorphic.
[Thus,
one
veri-
fies
immediately
that
composites
of
species-isomorphisms
are
species-isomorphisms.]
We
shall
refer
to
a
species-isomorphism
whose
domain
and
codomain
are
equal
as
a
species-automorphism.
We
shall
refer
to
as
model-free
[cf.
Remark
3.1.1
below]
an
i-specimen
of
S
equipped
with
a
description
via
a
set-theoretic
formula
that
is
“independent
of
the
ZFC-model
in
which
it
is
given”
in
the
sense
that
for
any
pair
of
universes
V
1
,
V
2
of
some
ZFC-model
such
that
V
1
∈
V
2
,
the
set-theoretic
formula
determines
the
same
i-specimen
of
S,
whether
interpreted
relative
to
the
ZFC-model
determined
by
V
1
or
the
ZFC-model
determined
by
V
2
.
70
SHINICHI
MOCHIZUKI
(iv)
We
shall
refer
to
as
the
category
determined
by
S
in
a
ZFC-model
the
category
whose
objects
are
the
species-objects
of
S
in
the
given
ZFC-model
and
whose
arrows
are
the
species-morphisms
of
S
in
the
given
ZFC-model.
[One
verifies
immediately
that
this
description
does
indeed
determine
a
category.]
Remark
3.1.1.
We
observe
that
any
of
the
familiar
descriptions
of
N
[cf.,
e.g.,
[Drk],
Chapter
2,
Definitions
2.3,
2.9],
Z,
Q,
Q
p
,
or
R,
for
instance,
yield
species
[all
of
whose
species-morphisms
are
identity
species-morphisms]
each
of
which
has
a
unique
species-object
in
any
given
ZFC-model.
Such
species
are
not
to
be
confused
with
such
species
as
the
species
of
“monoids
isomorphic
to
N
and
monoid
isomor-
phisms”,
which
admits
many
species-objects
[all
of
which
are
species-isomorphic]
in
any
ZFC-model.
On
the
other
hand,
the
set-theoretic
formula
used,
for
instance,
to
define
the
former
“species
N”
may
be
applied
to
define
a
“model-free
species-object
N”
of
the
latter
“species
of
monoids
isomorphic
to
N”.
Remark
3.1.2.
(i)
It
is
important
to
remember
when
working
with
species
that
the
essence
of
a
species
lies
not
in
the
specific
sets
that
occur
as
species-
objects
or
species-morphisms
of
the
species
in
various
ZFC-models,
but
rather
in
the
collection
of
rules,
i.e.,
set-theoretic
formulas,
that
gov-
ern
the
construction
of
such
sets
in
an
unspecified,
“indeterminate”
ZFC-
model.
Put
another
way,
the
emphasis
in
the
theory
of
species
lies
in
the
programs
—
i.e.,
“software”
—
that
yield
the
desired
output
data,
not
on
the
output
data
itself.
From
this
point
of
view,
one
way
to
describe
the
various
set-theoretic
formulas
that
constitute
a
species
is
as
a
“deterministic
algorithm”
[a
term
suggested
to
the
author
by
Minhyong
Kim]
for
constructing
the
sets
to
be
considered.
(ii)
One
interesting
point
of
view
that
arose
in
discussions
between
the
author
and
F.
Kato
is
the
following.
The
relationship
between
the
classical
approach
to
discussing
mathematics
relative
to
a
fixed
model
of
set
theory
—
an
approach
in
which
specific
sets
play
a
central
role
—
and
the
“species-theoretic”
approach
con-
sidered
here
—
in
which
the
rules,
given
by
set-theoretic
formulas
for
constructing
the
sets
of
interest
[i.e.,
not
specific
sets
themselves!],
play
a
central
role
—
may
be
regarded
as
analogous
to
the
relationship
between
classical
approaches
to
alge-
braic
varieties
—
in
which
specific
sets
of
solutions
of
polynomial
equations
in
an
algebraically
closed
field
play
a
central
role
—
and
scheme
theory
—
in
which
the
functor
determined
by
a
scheme,
i.e.,
the
polynomial
equations,
or
“rules”,
that
de-
termine
solutions,
as
opposed
to
specific
sets
of
solutions
themselves,
play
a
central
role.
That
is
to
say,
in
summary:
[fixed
model
of
set
theory
approach
:
species-theoretic
approach]
←→
[varieties
:
schemes]
A
similar
analogy
—
i.e.,
of
the
form
[fixed
model
of
set
theory
approach
:
species-theoretic
approach]
←→
[groups
of
specific
matrices
:
abstract
groups]
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
71
—
may
be
made
to
the
notion
of
an
“abstract
group”,
as
opposed
to
a
“group
of
specific
matrices”.
That
is
to
say,
just
as
a
“group
of
specific
matrices
may
be
thought
of
as
a
specific
representation
of
an
“abstract
group”,
the
category
of
objects
determined
by
a
species
in
a
specific
ZFC-model
may
be
thought
of
as
a
specific
representation
of
an
“abstract
species”.
(iii)
If,
in
the
context
of
the
discussion
of
(i),
(ii),
one
tries
to
form
a
sort
of
quotient,
in
which
“programs”
that
yield
the
same
sets
as
“output
data”
are
identified,
then
one
must
contend
with
the
resulting
indeterminacy,
i.e.,
working
with
programs
is
only
well-defined
up
to
internal
modifications
of
the
programs
in
question
that
does
not
affect
the
final
output.
This
leads
to
somewhat
intractable
problems
concerning
the
internal
structure
of
such
programs
—
a
topic
that
lies
well
beyond
the
scope
of
the
present
work.
Remark
3.1.3.
(i)
Typically,
in
the
discussion
to
follow,
we
shall
not
write
out
explicitly
the
various
set-theoretic
formulas
involved
in
the
definition
of
a
species.
Rather,
it
is
to
be
understood
that
the
set-theoretic
formulas
to
be
used
are
those
arising
from
the
conventional
descriptions
of
the
mathematical
objects
involved.
When
applying
such
conventional
descriptions,
however,
it
is
important
to
check
that
they
are
well-defined
and
do
not
depend
upon
the
use
of
arbitrary
choices
that
are
not
describable
via
well-defined
set-theoretic
formulas.
(ii)
The
fact
that
the
data
involved
in
a
species
is
given
by
abstract
set-theoretic
formulas
imparts
a
certain
canonicality
to
the
mathematical
notion
constituted
by
the
species,
a
canonicality
that
is
not
shared,
for
instance,
by
mathematical
objects
whose
construction
depends
on
an
invocation
of
the
axiom
of
choice
in
some
particular
ZFC-model
[cf.
the
discussion
of
(i)
above].
Moreover,
by
furnishing
a
stock
of
such
“canonical
notions”,
the
theory
of
species
allows
one,
in
effect,
to
compute
the
extent
of
deviation
of
various
“non-canonical
objects”
[i.e.,
whose
construction
depends
upon
the
invocation
of
the
axiom
of
choice!]
from
a
sort
of
“canonical
norm”.
Remark
3.1.4.
Note
that
because
the
data
involved
in
a
species
is
given
by
abstract
set-theoretic
formulas,
the
mathematical
notion
constituted
by
the
species
is
immune
to,
i.e.,
unaffected
by,
extensions
of
the
universe
—
i.e.,
such
as
the
ascending
chain
V
0
∈
V
1
∈
V
2
∈
V
3
∈
.
.
.
∈
V
n
∈
.
.
.
∈
V
that
appears
in
the
discussion
preceding
Definition
3.1
—
in
which
one
works.
This
is
the
sense
in
which
we
apply
the
term
“inter-universal”.
That
is
to
say,
“inter-universal
geometry”
allows
one
to
relate
the
“geometries”
that
occur
in
distinct
universes.
Remark
3.1.5.
Similar
remarks
to
the
remarks
made
in
Remarks
3.1.2,
3.1.3,
and
3.1.4
concerning
the
significance
of
working
with
set-theoretic
formulas
may
be
made
with
regard
to
the
notions
of
mutations,
morphisms
of
mutations,
mutation-
histories,
observables,
and
cores
to
be
introduced
in
Definition
3.3
below.
One
fundamental
example
of
a
species
is
the
following.
72
SHINICHI
MOCHIZUKI
Example
3.2.
Categories.
The
notions
of
a
[small]
category
and
an
isomor-
phism
class
of
[covariant]
functors
between
two
given
[small]
categories
yield
an
example
of
a
species.
That
is
to
say,
at
a
set-theoretic
level,
one
may
think
of
a
[small]
category
as,
for
instance,
a
set
of
arrows,
together
with
a
set
of
composition
relations,
that
satisfies
certain
properties;
one
may
think
of
a
[covariant]
functor
between
[small]
categories
as
the
set
given
by
the
graph
of
the
map
on
arrows
de-
termined
by
the
functor
[which
satisfies
certain
properties];
one
may
think
of
an
isomorphism
class
of
functors
as
a
collection
of
such
graphs,
i.e.,
the
graphs
deter-
mined
by
the
functors
in
the
isomorphism
class,
which
satisfies
certain
properties.
Then
one
has
“dictionaries”
0-species
1-species
←→
←→
the
notion
of
a
category
the
notion
of
an
isomorphism
class
of
functors
at
the
level
of
notions
and
a
0-specimen
a
1-specimen
←→
←→
a
particular
[small]
category
a
particular
isomorphism
class
of
functors
at
the
level
of
specific
mathematical
objects
in
a
specific
ZFC-model.
Moreover,
one
verifies
easily
that
species-isomorphisms
between
0-species
correspond
to
isomor-
phism
classes
of
equivalences
of
categories
in
the
usual
sense.
Remark
3.2.1.
Note
that
in
the
case
of
Example
3.2,
one
could
also
define
a
notion
of
“2-species”,
“2-specimens”,
etc.,
via
the
notion
of
an
“isomorphism
of
functors”,
and
then
take
the
1-species
under
consideration
to
be
the
notion
of
a
functor
[i.e.,
not
an
isomorphism
class
of
functors].
Indeed,
more
generally,
one
could
define
a
notion
of
“n-species”
for
arbitrary
integers
n
≥
1.
Since,
however,
this
approach
would
only
serve
to
add
an
unnecessary
level
of
complexity
to
the
theory,
we
choose
here
to
take
the
approach
of
working
with
“functors
considered
up
to
isomorphism”.
Definition
3.3.
Let
S
=
(S
0
,
S
1
);
S
=
(S
0
,
S
1
)
be
species.
(i)
A
mutation
M
:
S
S
is
defined
to
be
a
collection
of
set-theoretic
formulas
Ψ
0
,
Ψ
1
satisfying
the
following
properties:
(a)
Ψ
0
is
a
set-theoretic
formula
Ψ
0
(E,
E)
involving
a
collection
of
species-data
E
for
S
0
and
a
collection
of
species-
data
E
for
S
0
;
in
this
situation,
we
shall
write
M(E)
for
E.
Moreover,
if,
in
some
ZFC-model,
E
∈
S
0
,
then
we
require
that
there
exist
a
unique
E
∈
S
0
such
that
Ψ
0
(E,
E)
holds;
in
this
situation,
we
shall
write
M(E)
for
E.
(b)
Ψ
1
is
a
set-theoretic
formula
Ψ
1
(E,
E
,
F,
F)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
73
involving
a
collection
of
species-data
F
:
E
→
E
for
S
1
and
a
collection
of
species-data
F
:
E
→
E
for
S
1
,
where
E
=
M(E),
E
=
M(E
);
in
this
situation,
we
shall
write
M(F)
for
F.
Moreover,
if,
in
some
ZFC-
model,
(F
:
E
→
E
)
∈
S
1
,
then
we
require
that
there
exist
a
unique
(F
:
E
→
E
)
∈
S
1
such
that
Ψ
0
(E,
E
,
F,
F
)
holds;
in
this
situation,
we
shall
write
M(F
)
for
F
.
Finally,
we
require
that
the
assignment
F
→
M(F
)
be
compatible
with
composites
and
map
identity
species-morphisms
of
S
to
identity
species-morphisms
of
S.
In
particular,
if
one
fixes
a
ZFC-
model,
then
M
determines
a
functor
from
the
category
determined
by
S
in
the
given
ZFC-model
to
the
category
determined
by
S
in
the
given
ZFC-model.
There
are
evident
notions
of
“composition
of
mutations”
and
“identity
mutations”.
(ii)
Let
M,
M
:
S
S
be
mutations.
Then
a
morphism
of
mutations
Z
:
M
→
M
is
defined
to
be
a
set-theoretic
formula
Ξ
satisfying
the
following
properties:
(a)
Ξ
is
a
set-theoretic
formula
Ξ(E,
F)
involving
a
collection
of
species-data
E
for
S
0
and
a
collection
of
species-
data
F
:
M(E)
→
M
(E)
for
S
1
;
in
this
situation,
we
shall
write
Z(E)
for
F.
Moreover,
if,
in
some
ZFC-model,
E
∈
S
0
,
then
we
require
that
there
exist
a
unique
F
∈
S
1
such
that
Ξ(E,
F
)
holds;
in
this
situation,
we
shall
write
Z(E)
for
F
.
(b)
Suppose,
in
some
ZFC-model,
that
F
:
E
1
→
E
2
is
a
species-morphism
of
S.
Then
one
has
an
equality
of
composite
species-morphisms
M
(F
)
◦
Z(E
1
)
=
Z(E
2
)
◦
M(F
)
:
M(E
1
)
→
M
(E
2
).
In
particular,
if
one
fixes
a
ZFC-model,
then
a
morphism
of
mutations
M
→
M
determines
a
natural
transformation
between
the
functors
determined
by
M,
M
in
the
ZFC-
model
—
cf.
(i).
There
are
evident
notions
of
“composition
of
morphisms
of
mutations”
and
“identity
morphisms
of
mutations”.
If
it
holds
that
for
every
species-object
E
of
S,
Z(E)
is
a
species-isomorphism,
then
we
shall
refer
to
Z
as
an
isomorphism
of
mutations.
In
particular,
one
verifies
immediately
that
Z
is
an
isomorphism
of
mutations
if
and
only
if
there
exists
a
morphism
of
mutations
Z
:
M
→
M
such
that
the
composite
morphisms
of
mutations
Z
◦
Z
:
M
→
M,
Z
◦
Z
:
M
→
M
are
the
respective
identity
morphisms
of
the
mutations
M,
M
.
(iii)
Let
M
:
S
S
be
a
mutation.
Then
we
shall
say
that
M
is
a
mutation-
equivalence
if
there
exists
a
mutation
M
:
S
S,
together
with
isomorphisms
of
mutations
between
the
composites
M
◦
M
,
M
◦
M
and
the
respective
iden-
tity
mutations.
In
this
situation,
we
shall
say
that
M,
M
are
mutation-quasi-
inverses
to
one
another.
Finally,
we
observe
that,
if
we
suppose
further
that
S,
S
are
1-small,
then
for
any
two
given
species-objects
in
the
domain
species
of
a
mutation-equivalence,
the
mutation-equivalence
induces
a
bijection
between
the
set
of
species-morphisms
(respectively,
species-isomorphisms)
between
the
two
74
SHINICHI
MOCHIZUKI
given
species-objects
[of
the
domain
species]
and
the
set
of
species-morphisms
(re-
spectively,
species-isomorphisms)
between
the
two
species-objects
[of
the
codomain
species]
obtained
by
applying
the
mutation-equivalence
to
the
two
given
species-
objects.
(iv)
Let
Γ
be
an
oriented
graph,
i.e.,
a
graph
Γ,
which
we
shall
refer
to
as
the
underlying
graph
of
Γ,
equipped
with
the
additional
data
of
a
total
ordering,
for
each
edge
e
of
Γ,
on
the
set
[of
cardinality
2]
of
branches
of
e
[cf.,
e.g.,
[AbsTopIII],
§0].
Then
we
define
a
mutation-history
H
=
(
Γ,
S
∗
,
M
∗
)
[indexed
by
Γ]
to
be
a
collection
of
data
as
follows:
(a)
for
each
vertex
v
of
Γ,
a
species
S
v
;
(b)
for
each
edge
e
of
Γ,
running
from
a
vertex
v
1
to
a
vertex
v
2
,
a
mutation
M
e
:
S
v
1
S
v
2
.
In
this
situation,
we
shall
refer
to
the
vertices,
edges,
and
branches
of
Γ
as
vertices,
edges,
and
branches
of
H.
Thus,
the
notion
of
a
“mutation-history”
may
be
thought
of
as
a
species-theoretic
version
of
the
notion
of
a
“diagram
of
categories”
given
in
[AbsTopIII],
Definition
3.5,
(i).
(v)
Let
H
=
(
Γ,
S
∗
,
M
∗
)
be
a
mutation-history;
S
a
species.
For
simplicity,
we
assume
that
the
underlying
graph
of
Γ
is
simply
connected.
Then
we
shall
refer
to
as
a(n)
[S-valued]
covariant
(respectively,
contravariant)
observable
V
of
the
mutation-history
H
a
collection
of
data
as
follows:
(a)
for
each
vertex
v
of
Γ,
a
mutation
V
v
:
S
v
→
S,
which
we
shall
refer
to
as
the
observation
mutation
at
v;
(b)
for
each
edge
e
of
Γ,
running
from
a
vertex
v
1
to
a
vertex
v
2
,
a
morphism
of
mutations
V
e
:
V
v
1
→
V
v
2
◦
M
e
(respectively,
V
e
:
V
v
2
◦
M
e
→
V
v
1
).
If
V
is
a
covariant
observable
such
that
all
of
the
morphisms
of
mutations
“V
e
”
are
isomorphisms
of
mutations,
then
we
shall
refer
to
the
covariant
observable
V
as
a
core.
Thus,
one
may
think
of
a
core
C
of
a
mutation-history
as
lying
“under”
the
entire
mutation-history
in
a
“uniform
fashion”.
Also,
we
shall
refer
to
the
“property
[of
an
observable]
of
being
a
core”
as
the
“coricity”
of
the
observable.
Finally,
we
note
that
the
notions
of
an
“observable”
and
a
“core”
given
here
may
be
thought
of
as
simplified,
species-theoretic
versions
of
the
notions
of
“observable”
and
“core”
given
in
[AbsTopIII],
Definition
3.5,
(iii).
Remark
3.3.1.
(i)
One
well-known
consequence
of
the
axiom
of
foundation
of
axiomatic
set
theory
is
the
assertion
that
“∈-loops”
a
∈
b
∈
c
∈
...
∈
a
can
never
occur
in
the
set
theory
in
which
one
works.
On
the
other
hand,
there
are
many
situations
in
mathematics
in
which
one
wishes
to
somehow
“identify”
mathematical
objects
that
arise
at
higher
levels
of
the
∈-structure
of
the
set
theory
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
75
under
consideration
with
mathematical
objects
that
arise
at
lower
levels
of
this
∈-structure.
In
some
sense,
the
notions
of
a
“set”
and
of
a
“bijection
of
sets”
allow
one
to
achieve
such
“identifications”.
That
is
to
say,
the
mathematical
objects
at
both
higher
and
lower
levels
of
the
∈-structure
constitute
examples
of
the
same
mathematical
notion
of
a
“set”,
so
that
one
may
consider
“bijections
of
sets”
be-
tween
those
sets
without
violating
the
axiom
of
foundation.
In
some
sense,
the
notion
of
a
species
may
be
thought
of
as
a
natural
extension
of
this
observation.
That
is
to
say,
the
notion
of
a
“species”
allows
one
to
consider,
for
instance,
species-
isomorphisms
between
species-objects
that
occur
at
different
levels
of
the
∈-structure
of
the
set
theory
under
consideration
—
i.e.,
roughly
speaking,
to
“simulate
∈-loops”
—
without
violating
the
axiom
of
foundation.
Moreover,
typically
the
sorts
of
species-objects
at
different
levels
of
the
∈-structure
that
one
wishes
to
somehow
have
“identified”
with
one
another
occur
as
the
result
of
executing
the
mutations
that
arise
in
some
sort
of
mutation-history
...
S
S
S
...
S
...
[where
S
=
(S
0
,
S
1
);
S
=
(S
0
,
S
1
);
S
=
(S
0
,
S
1
)
are
species]
—
e.g.,
the
“output
species-objects”
of
the
“S”
on
the
right
that
arise
from
applying
various
mutations
to
the
“input
species-objects”
of
the
“S”
on
the
left.
(ii)
In
the
context
of
constructing
“loops”
in
a
mutation-history
as
in
the
final
display
of
(i),
we
observe
that
the
simpler
the
structure
of
the
species
involved,
the
easier
it
is
to
construct
“loops”.
It
is
for
this
reason
that
species
such
as
the
species
determined
by
the
notion
of
a
category
[cf.
Example
3.2]
are
easier
to
work
with,
from
the
point
of
view
of
constructing
“loops”,
than
more
complicated
species
such
as
the
species
determined
by
the
notion
of
a
scheme.
This
is
one
of
the
principal
motivations
for
the
“geometry
of
categories”
—
of
which
“absolute
anabelian
geometry”
is
the
special
case
that
arises
when
the
categories
involved
are
Galois
categories
—
i.e.,
for
the
theory
of
representing
scheme-theoretic
geometries
via
categories
[cf.,
e.g.,
the
Introductions
of
[MnLg],
[SemiAnbd],
[Cusp],
[FrdI]].
At
a
more
concrete
level,
the
utility
of
working
with
categories
to
reconstruct
objects
that
occurred
at
earlier
stages
of
some
sort
of
“series
of
constructions”
[cf.
the
mutation-history
of
the
final
display
of
(i)!]
may
be
seen
in
the
“reconstruction
of
the
underlying
scheme”
in
various
situations
throughout
[MnLg]
by
applying
the
natural
equivalence
of
categories
of
the
final
display
of
[MnLg],
Definition
1.1,
(iv),
from
a
certain
category
constructed
from
a
log
scheme,
as
well
as
in
the
theory
of
“slim
exponentiation”
discussed
in
the
Appendix
to
[FrdI].
(iii)
Again
in
the
context
of
mutation-histories
such
as
the
one
given
in
the
final
display
of
(i),
although
one
may,
on
certain
occasions,
wish
to
apply
various
mutations
that
fundamentally
alter
the
structure
of
the
mathematical
objects
in-
volved
and
hence
give
rise
to
“output
species-objects”
of
the
“S”
on
the
right
that
are
related
in
a
highly
nontrivial
fashion
to
the
“input
species-objects”
of
the
“S”
on
the
left,
it
is
also
of
interest
to
consider
76
SHINICHI
MOCHIZUKI
“portions”
of
the
various
mathematical
objects
that
occur
that
are
left
unaltered
by
the
various
mutations
that
one
applies.
This
is
precisely
the
reason
for
the
introduction
of
the
notion
of
a
core
of
a
mutation-
history.
One
important
consequence
of
the
construction
of
various
cores
associated
to
a
mutation-history
is
that
often
one
may
apply
various
cores
associated
to
a
mutation-history
to
describe,
by
means
of
non-coric
observables,
the
portions
of
the
various
math-
ematical
objects
that
occur
which
are
altered
by
the
various
mutations
that
one
applies
in
terms
of
the
unaltered
portions,
i.e.,
cores.
Indeed,
this
point
of
view
plays
a
central
role
in
the
theory
of
the
present
series
of
papers
—
cf.
the
discussion
of
Remark
3.6.1,
(ii),
below.
Remark
3.3.2.
One
somewhat
naive
point
of
view
that
constituted
one
of
the
original
motivations
for
the
author
in
the
development
of
theory
of
the
present
series
of
papers
is
the
following.
In
the
classical
theory
of
schemes,
when
considering
local
systems
on
a
scheme,
there
is
no
reason
to
restrict
oneself
to
considering
local
systems
valued
in,
say,
modules
over
a
finite
ring.
If,
moreover,
there
is
no
reason
to
make
such
a
restriction,
then
one
is
naturally
led
to
consider,
for
instance,
local
systems
of
schemes
[cf.,
e.g.,
the
theory
of
the
“Galois
mantle”
in
[pTeich]],
or,
indeed,
local
systems
of
more
general
collections
of
mathematical
objects.
One
may
then
ask
what
happens
if
one
tries
to
consider
local
systems
on
the
schemes
that
occur
as
fibers
of
a
local
system
of
schemes.
[More
concretely,
if
X
is,
for
instance,
a
connected
scheme,
then
one
may
consider
local
systems
X
over
X
whose
fibers
are
isomorphic
to
X;
then
one
may
repeat
this
process,
by
considering
such
local
systems
over
each
fiber
of
the
local
system
X
on
X,
etc.]
In
this
way,
one
is
eventually
led
to
the
consideration
of
“systems
of
nested
local
systems”
—
i.e.,
a
local
system
over
a
local
system
over
a
local
system,
etc.
It
is
precisely
this
point
of
view
that
underlies
the
notion
of
“successive
iteration
of
a
given
mutation-history”,
relative
to
the
terminology
formulated
in
the
present
§3.
If,
moreover,
one
thinks
of
such
“successive
iterates
of
a
given
mutation-history”
as
being
a
sort
of
abstraction
of
the
naive
idea
of
a
“system
of
nested
local
systems”,
then
the
notion
of
a
core
may
be
thought
of
as
a
sort
of
mathematical
object
that
is
invariant
with
respect
to
the
application
of
the
operations
that
gave
rise
to
the
“system
of
nested
local
systems”.
Example
3.4.
Topological
Spaces
and
Fundamental
Groups.
(i)
One
verifies
easily
that
the
notions
of
a
topological
space
and
a
continuous
map
between
topological
spaces
determine
an
example
of
a
species
S
top
.
In
a
similar
→
X
of
a
pathwise
connected
topological
vein,
the
notions
of
a
universal
covering
X
→
X,
Y
→
Y
space
X
and
a
continuous
map
between
such
universal
coverings
X
→
Y
,
X
→
Y
],
considered
up
to
[i.e.,
a
pair
of
compatible
continuous
maps
X
composition
with
a
deck
transformation
of
the
universal
covering
Y
→
Y
,
determine
an
example
of
a
species
S
u-top
.
We
leave
to
the
reader
the
routine
task
of
writing
out
the
various
set-theoretic
formulas
that
define
the
species
structures
of
S
top
,
S
u-top
.
Here,
we
note
that
at
a
set-theoretic
level,
the
species-morphisms
of
S
u-top
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
77
are
collections
of
continuous
maps
[between
two
given
universal
coverings],
any
two
of
which
differ
from
one
another
by
composition
with
a
deck
transformation.
(ii)
One
verifies
easily
that
the
notions
of
a
group
and
an
outer
homomorphism
between
groups
[i.e.,
a
homomorphism
considered
up
to
composition
with
an
inner
automorphism
of
the
codomain
group]
determine
an
example
of
a
species
S
gp
.
We
leave
to
the
reader
the
routine
task
of
writing
out
the
various
set-theoretic
formulas
that
define
the
species
structure
of
S
gp
.
Here,
we
note
that
at
a
set-theoretic
level,
the
species-morphisms
of
S
gp
are
collections
of
homomorphisms
[between
two
given
groups],
any
two
of
which
differ
from
one
another
by
composition
with
an
inner
automorphism.
(iii)
Now
one
verifies
easily
that
the
assignment
→
X)
(
X
→
Aut(
X/X)
→
X)
is
a
species-object
of
S
u-top
,
and
Aut(
X/X)
denotes
the
group
—
where
(
X
of
deck
transformations
of
the
universal
covering
X
→
X
—
determines
a
mutation
S
u-top
S
gp
.
That
is
to
say,
the
“fundamental
group”
may
be
thought
of
as
a
sort
of
mutation.
Example
3.5.
Absolute
Anabelian
Geometry.
(i)
Let
S
be
a
class
of
connected
normal
schemes
that
is
closed
under
isomor-
phism
[of
schemes].
Suppose
that
there
exists
a
set
E
S
of
schemes
describable
by
a
set-theoretic
formula
with
the
property
that
every
scheme
of
S
is
isomorphic
to
some
scheme
belonging
to
E
S
.
Then
just
as
in
the
case
of
universal
coverings
of
topological
spaces
discussed
in
Example
3.4,
(i),
one
verifies
easily,
by
applying
the
set-theoretic
formula
describing
E
S
,
that
the
universal
pro-finite
étale
cover-
→
X
of
schemes
X
belonging
to
S
and
isomorphisms
of
such
coverings
ings
X
considered
up
to
composition
with
a
deck
transformation
give
rise
to
a
species
S
S
.
(ii)
Let
G
be
a
class
of
topological
groups
that
is
closed
under
isomorphism
[of
topological
groups].
Suppose
that
there
exists
a
set
E
G
of
topological
groups
describable
by
a
set-theoretic
formula
with
the
property
that
every
topological
group
of
G
is
isomorphic
to
some
topological
group
belonging
to
E
G
.
Then
just
as
in
the
case
of
abstract
groups
discussed
in
Example
3.4,
(ii),
one
verifies
easily,
by
applying
the
set-theoretic
formula
describing
E
G
,
that
topological
groups
belonging
to
G
and
[bi-continuous]
outer
isomorphisms
between
such
topological
groups
give
rise
to
a
species
S
G
.
(iii)
Let
S
be
as
in
(i).
Then
for
an
appropriate
choice
of
G,
by
associating
to
a
universal
pro-finite
étale
covering
the
resulting
group
of
deck
transformations,
one
obtains
a
mutation
Π
:
S
S
S
G
[cf.
Example
3.4,
(iii)].
Then
one
way
to
define
the
notion
that
the
schemes
belonging
to
the
class
S
are
“[absolute]
anabelian”
is
to
require
the
specification
of
a
mutation
S
S
A
:
S
G
78
SHINICHI
MOCHIZUKI
which
forms
a
mutation-quasi-inverse
to
Π.
Here,
we
note
that
the
existence
of
the
bijections
[i.e.,
“fully
faithfulness”]
discussed
in
Definition
3.3,
(iii),
is,
in
essence,
the
condition
that
is
usually
taken
as
the
definition
of
“anabelian”.
By
contrast,
the
species-theoretic
approach
of
the
present
discussion
may
be
thought
of
as
an
explicit
mathematical
formulation
of
the
algorithmic
approach
to
[absolute]
an-
abelian
geometry
discussed
in
the
Introduction
to
[AbsTopI].
(iv)
The
framework
of
[absolute]
anabelian
geometry
[cf.,
e.g.,
the
framework
discussed
above
in
(iii)]
gives
a
good
example
of
the
importance
of
specifying
pre-
cisely
what
species
one
is
working
with
in
a
given
“series
of
constructions”
[cf.,
e.g.,
the
mutation-history
of
the
final
display
of
Remark
3.3.1,
(i)].
That
is
to
say,
there
is
a
quite
substantial
difference
between
working
with
a
profinite
group
in
its
sole
capacity
as
a
profinite
group
and
working
with
the
same
profinite
group
—
which
may
happen
to
arise
as
the
étale
fundamental
group
of
some
scheme!
—
regarded
as
being
equipped
with
various
data
that
arise
from
the
construc-
tion
of
the
profinite
group
as
the
étale
fundamental
group
of
some
scheme.
It
is
precisely
this
sort
of
issue
that
constituted
one
of
the
original
motivations
for
the
author
in
the
development
of
the
theory
of
species
presented
here.
Example
3.6.
The
Étale
Site
and
Frobenius.
(i)
Let
p
be
a
prime
number.
If
S
is
a
reduced
scheme
over
F
p
,
then
denote
by
S
the
scheme
with
the
same
topological
space
as
S,
but
whose
structure
sheaf
is
given
by
the
subsheaf
def
O
S
(p)
=
(O
S
)
p
⊆
O
S
(p)
of
p-th
powers
of
sections
of
S.
Thus,
the
natural
inclusion
O
S
(p)
→
O
S
induces
a
morphism
Φ
S
:
S
→
S
(p)
.
Moreover,
“raising
to
the
p-th
power”
determines
a
∼
natural
isomorphism
α
S
:
S
(p)
→
S
such
that
the
resulting
composite
α
S
◦
Φ
S
:
S
→
S
is
the
Frobenius
morphism
of
S.
Write
S
p-sch
for
the
species
of
reduced
quasi-compact
schemes
over
F
p
and
quasi-compact
mor-
phisms
of
schemes.
Then
consider
the
[small]
category
S
ét
—
i.e.,
“the
small
étale
site
of
S”
—
defined
as
follows:
An
object
of
S
ét
is
a(n)
[necessarily
quasi-affine,
by
Zariski’s
Main
Theo-
rem!]
étale
morphism
of
finite
presentation
T
→
S
equipped
with
a
finite
open
cover
{U
i
}
i∈I
of
S,
together
with
factorizations
i
T
|
U
i
⊆
A
N
U
i
→
U
i
for
each
i
∈
I
i
—
where
I
is
a
finite
subset
of
the
set
of
open
subschemes
of
S;
A
N
U
i
denotes
a
standard
copy
of
affine
N
i
-space
over
U
i
,
for
some
integer
N
i
≥
1;
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
79
i
“⊆”
exhibits
T
|
U
i
as
a
finitely
presented
subscheme
of
A
N
U
i
;
we
observe
that
any
étale
morphism
of
finite
presentation
T
→
S
necessarily
admits
such
auxiliary
data
parametrized
by
some
index
set
I.
A
morphism
of
S
ét
from
an
object
T
1
→
S
to
an
object
T
2
→
S
[each
of
which
is
equipped
with
auxiliary
data]
is
a(n)
[necessarily
étale
of
finite
presentation]
S-morphism
T
1
→
T
2
.
In
particular,
one
may
construct
an
assignment
S
→
S
ét
that
maps
a
species-object
S
of
S
p-sch
to
the
[small]
category
S
ét
in
such
a
way
that
the
assignment
S
→
S
ét
is
contravariantly
functorial
with
respect
to
species-
morphisms
S
1
→
S
2
of
S
p-sch
,
and,
moreover,
may
be
described
via
set-theoretic
formulas.
Thus,
such
an
assignment
determines
an
“étale
site
mutation”
M
ét
:
S
p-sch
S
cat
—
where
we
write
S
cat
for
the
species
of
categories
and
isomorphism
classes
of
contravariant
functors
[i.e.,
a
slightly
modified
form
of
the
species
considered
in
Example
3.2].
Another
natural
assignment
in
the
present
context
is
the
assignment
S
→
S
pf
which
maps
S
to
its
perfection
S
pf
,
i.e.,
the
scheme
determined
by
taking
the
inverse
limit
of
the
inverse
system
.
.
.
→
S
→
S
→
S
obtained
by
considering
iterates
of
the
Frobenius
morphism
of
S.
Thus,
by
considering
the
final
copy
of
“S”
in
this
inverse
system,
one
obtains
a
natural
morphism
β
S
:
S
pf
→
S.
Finally,
one
obtains
a
“perfection
mutation”
M
pf
:
S
p-sch
S
p-sch
by
considering
the
set-theoretic
formulas
underlying
the
assignment
S
→
S
pf
.
(ii)
Write
F
p-sch
:
S
p-sch
S
p-sch
for
the
“Frobenius
mutation”
obtained
by
considering
the
set-theoretic
formulas
underlying
the
assignment
S
→
S
(p)
.
Thus,
one
may
formulate
the
well-known
“invariance
of
the
étale
site
under
Frobenius”
[cf.,
e.g.,
[FK],
Chapter
I,
Proposition
3.16]
as
the
statement
that
the
“étale
site
mutation”
M
ét
exhibits
S
cat
as
a
core
—
i.e.,
an
“invariant
piece”
—
of
the
“Frobenius
mutation-history”
.
.
.
S
p-sch
S
p-sch
S
p-sch
S
p-sch
.
.
.
determined
by
the
“Frobenius
mutation”
F
p-sch
.
In
this
context,
we
observe
that
the
“perfection
mutation”
M
pf
also
yields
a
core
—
i.e.,
another
“invari-
ant
piece”
—
of
the
Frobenius
mutation-history.
On
the
other
hand,
the
nat-
ural
morphism
Φ
S
:
S
→
S
(p)
may
be
interpreted
as
a
covariant
observable
of
this
mutation-history
whose
observation
mutations
are
the
identity
mutations
id
S
p-sch
:
S
p-sch
S
p-sch
.
Since
Φ
S
is
not,
in
general,
an
isomorphism,
it
follows
80
SHINICHI
MOCHIZUKI
that
this
observable
constitutes
an
example
of
an
non-coric
observable.
Never-
theless,
the
natural
morphism
β
S
:
S
pf
→
S
may
be
interpreted
as
a
morphism
of
mutations
M
pf
→
id
S
p-sch
that
serves
to
relate
the
non-coric
observable
just
considered
to
the
coric
observable
arising
from
M
pf
.
(iii)
One
may
also
develop
a
version
of
(i),
(ii)
for
log
schemes;
we
leave
the
routine
details
to
the
interested
reader.
Here,
we
pause
to
mention
that
the
theory
of
log
schemes
motivates
the
following
“combinatorial
monoid-theoretic”
version
of
the
non-coric
observable
on
the
Frobenius
mutation-history
of
(ii).
Write
S
mon
for
the
species
of
torsion-free
abelian
monoids
and
morphisms
of
monoids.
If
M
def
is
a
species-object
of
S
mon
,
then
write
M
(p)
=
p
·
M
⊆
M
.
Then
the
assignment
M
→
M
(p)
determines
a
“monoid-Frobenius
mutation”
F
mon
:
S
mon
S
mon
and
hence
a
“monoid-Frobenius
mutation-history”
.
.
.
S
mon
S
mon
.
.
.
which
is
equipped
with
a
non-coric
contravariant
observable
determined
by
the
natural
inclusion
morphism
M
(p)
→
M
and
the
observation
mutations
given
by
the
identity
mutations
id
S
mon
:
S
mon
S
mon
.
On
the
other
hand,
the
p-perfection
M
pf
of
M
,
i.e.,
the
inductive
limit
of
the
inductive
system
M
→
M
→
M
→
.
.
.
obtained
by
considering
the
inclusions
given
by
multiplying
by
p,
gives
rise
to
a
“monoid-p-perfection
mutation”
M
pf-mon
:
S
mon
S
mon
—
which
may
be
interpreted
as
a
core
of
the
monoid-Frobenius
mutation-history.
Finally,
the
natural
inclusion
of
monoids
M
→
M
pf
may
be
interpreted
as
a
mor-
phism
of
mutations
id
S
mon
→
M
pf-mon
that
serves
to
relate
the
non-coric
observable
just
considered
to
the
coric
observable
arising
from
M
pf-mon
.
Remark
3.6.1.
(i)
The
various
constructions
of
Example
3.6
may
be
thought
of
as
providing,
in
the
case
of
the
phenomena
of
“invariance
of
the
étale
site
under
Frobenius”
and
“invariance
of
the
perfection
under
Frobenius”,
a
“species-theoretic
interpretation”
—
i.e.,
via
consideration
of
“coric”
versus
“non-coric”
observables
—
of
the
difference
between
“étale-type”
and
“Frobenius-type”
structures
[cf.
the
discussion
of
[FrdI],
§I4].
This
sort
of
approach
via
“combinatorial
patterns”
to
expressing
the
difference
between
“étale-type”
and
“Frobenius-type”
structures
plays
a
central
role
in
the
theory
of
the
present
series
of
papers.
Indeed,
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
81
mutation-histories
and
cores
considered
in
Example
3.6,
(ii),
(iii),
may
be
thought
of
as
the
underlying
motivating
examples
for
the
theory
of
both
·
the
vertical
lines,
i.e.,
consisting
of
log-links,
and
×μ
×μ
·
the
horizontal
lines,
i.e.,
consisting
of
Θ
×μ
-/Θ
×μ
gau
-/Θ
LGP
-/Θ
lgp
-links,
of
the
log-theta-lattice
[cf.
[IUTchIII],
Definitions
1.4,
3.8].
Finally,
we
recall
that
this
approach
to
understanding
the
log-links
may
be
seen
in
the
introduction
of
the
terminology
of
“observables”
and
“cores”
in
[AbsTopIII],
Definition
3.5,
(iii).
(ii)
Example
3.6
also
provides
a
good
example
of
the
important
theme
[cf.
the
discussion
of
Remark
3.3.1,
(iii)]
of
describing
non-coric
data
in
terms
of
coric
data
—
cf.
the
morphism
β
S
:
S
pf
→
S
of
Example
3.6,
(ii);
the
natural
inclusion
M
→
M
pf
of
Example
3.6,
(iii).
From
the
point
of
view
of
the
vertical
and
hori-
zontal
lines
of
the
log-theta-lattice
[cf.
the
discussion
of
(i)],
this
theme
may
also
be
observed
in
the
vertically
coric
log-shells
that
serve
as
a
common
receptacle
for
the
various
arrows
of
the
log-Kummer
correspondences
of
[IUTchIII],
Theorem
3.11,
(ii),
as
well
as
in
the
multiradial
representations
of
[IUTchIII],
Theorem
3.11,
(i),
which
describe
[certain
aspects
of]
the
arithmetic
holomorphic
structure
on
one
vertical
line
of
the
log-theta-lattice
in
terms
that
may
be
understood
rela-
tive
to
an
alien
arithmetic
holomorphic
structure
on
another
vertical
line
—
i.e.,
separated
from
the
first
vertical
line
by
horizontal
arrows
—
of
the
log-theta-lattice
[cf.
[IUTchIII],
Remark
3.11.1;
[IUTchIII],
Remark
3.12.2,
(ii)].
Remark
3.6.2.
(i)
In
the
context
of
the
theme
of
“coric
descriptions
of
non-coric
data”
dis-
cussed
in
Remark
3.6.1,
(ii),
it
is
of
interest
to
observe
the
significance
of
the
use
of
set-theoretic
formulas
[cf.
the
discussion
of
Remarks
3.1.2,
3.1.3,
3.1.4,
3.1.5]
to
realize
such
descriptions.
That
is
to
say,
descriptions
in
terms
of
arbitrary
choices
that
depend
on
a
particular
model
of
set
theory
[cf.
Remark
3.1.3]
do
not
allow
one
to
calculate
in
terms
that
make
sense
in
one
universe
the
operations
performed
in
an
alien
universe!
This
is
precisely
the
sort
of
situation
that
one
encoun-
ters
when
one
considers
the
vertical
and
horizontal
arrows
of
the
log-theta-lattice
[cf.
(ii)
below],
where
distinct
universes
arise
from
the
distinct
scheme-theoretic
basepoints
on
either
side
of
such
an
arrow
that
correspond
to
distinct
ring
the-
ories,
i.e.,
ring
theories
that
cannot
be
related
to
one
another
by
means
of
a
ring
homomorphism
—
cf.
the
discussion
of
Remark
3.6.3
below.
Indeed,
it
was
precisely
the
need
to
understand
this
sort
of
situation
that
led
the
author
to
develop
the
“inter-universal”
version
of
Teichmüller
theory
exposed
in
the
present
series
of
papers.
Finally,
we
observe
that
the
algorithmic
approach
[i.e.,
as
opposed
to
the
“fully
faithfulness/Grothendieck
Conjecture-style
approach”
—
cf.
Example
3.5,
(iii)]
to
reconstruction
issues
via
set-theoretic
formulas
plays
an
essential
role
in
this
con-
text.
That
is
to
say,
although
different
algorithms,
or
software,
may
yield
the
82
SHINICHI
MOCHIZUKI
same
output
data,
it
is
only
by
working
with
specific
algorithms
that
one
may
understand
the
delicate
inter-relations
that
exist
between
various
components
of
the
structures
that
occur
as
one
performs
various
operations
[i.e.,
the
mutations
of
a
mutation-history].
In
the
case
of
the
theory
developed
in
the
present
series
of
papers,
one
central
example
of
this
phenomenon
is
the
cyclotomic
rigidity
isomorphisms
that
underlie
the
theory
of
Θ
×μ
LGP
-link
compatibility
discussed
in
[IUTchIII],
Theorem
3.11,
(iii),
(c),
(d)
[cf.
also
[IUTchIII],
Remarks
2.2.1,
2.3.2].
(ii)
The
algorithmic
approach
to
reconstruction
that
is
taken
throughout
the
present
series
of
papers,
as
well
as,
for
instance,
in
[FrdI],
[EtTh],
and
[AbsTopIII],
was
conceived
by
the
author
in
the
spirit
of
the
species-theoretic
formulation
ex-
posed
in
the
present
§3.
Nevertheless,
[cf.
Remark
3.1.3,
(i)]
we
shall
not
explicitly
write
out
the
various
set-theoretic
formulas
involved
in
the
various
species,
muta-
tions,
etc.
that
are
implicit
throughout
the
theory
of
the
present
series
of
papers.
Rather,
it
is
to
be
understood
that
the
set-theoretic
formulas
to
be
used
are
those
arising
from
the
conventional
descriptions
that
are
given
of
the
mathematical
ob-
jects
involved.
When
applying
such
conventional
descriptions,
however,
the
reader
is
obliged
to
check
that
they
are
well-defined
and
do
not
depend
upon
the
use
of
arbitrary
choices
that
are
not
describable
via
well-defined
set-theoretic
formulas.
(iii)
The
sharp
contrast
between
·
the
canonicality
imparted
by
descriptions
via
set-theoretic
formulas
in
the
context
of
extensions
of
the
universe
in
which
one
works
[cf.
Remarks
3.1.3,
3.1.4]
and
·
the
situation
that
arises
if
one
allows,
in
one’s
descriptions,
the
various
arbitrary
choices
arising
from
invocations
of
the
axiom
of
choice
may
be
understood
somewhat
explicitly
if
one
attempts
to
“catalogue
the
various
possibilities”
corresponding
to
various
possible
choices
that
may
occur
in
one’s
de-
scription.
That
is
to
say,
such
a
“cataloguing
operation”
typically
obligates
one
to
work
with
“sets
of
very
large
cardinality”,
many
of
which
must
be
constructed
by
means
of
set-theoretic
exponentiation
[i.e.,
such
as
the
operation
of
passing
from
a
set
E
to
the
power
set
“2
E
”
of
all
subsets
of
E].
Such
a
rapid
outbreak
of
“unwieldy
large
sets”
is
reminiscent
of
the
rapid
growth,
in
the
p-adic
crystalline
theory,
of
the
p-adic
valuations
of
the
denominators
that
occur
when
one
formally
integrates
an
arbitrary
connection,
as
opposed
to
a
“canonical
connection”
of
the
sort
that
arises
from
a
crystalline
representation.
In
the
p-adic
theory,
such
“canonical
connections”
are
typically
related
to
“canonical
liftings”,
such
as,
for
instance,
those
that
occur
in
p-adic
Teichmüller
theory
[cf.
[pOrd],
[pTeich]].
In
this
context,
it
is
of
interest
to
recall
that
the
canonical
liftings
of
p-adic
Teichmüller
theory
may,
under
certain
conditions,
be
thought
of
as
liftings
“of
minimal
com-
plexity”
in
the
sense
that
their
Witt
vector
coordinates
are
given
by
polynomials
of
minimal
degree
—
cf.
the
computations
of
[Finot].
Remark
3.6.3.
(i)
In
the
context
of
Remark
3.6.2,
it
is
useful
to
recall
the
fundamental
reason
for
the
need
to
pursue
“inter-universality”
in
the
present
series
of
papers
[cf.
the
discussion
of
[IUTchIII],
Remark
1.2.4;
[IUTchIII],
Remark
1.4.2],
namely,
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
83
since
étale
fundamental
groups
—
i.e.,
in
essence,
Galois
groups
—
are
defined
as
certain
automorphism
groups
of
fields/rings,
the
definition
of
such
a
Galois
group
as
a
certain
automorphism
group
of
some
ring
struc-
ture
is
fundamentally
incompatible
with
the
vertical
and
horizontal
arrows
of
the
log-theta-lattice
[i.e.,
which
do
not
arise
from
ring
homo-
morphisms]!
In
this
respect,
“transformations”
such
as
the
vertical
and
horizontal
arrows
of
the
log-theta-lattice
differ,
quite
fundamentally,
from
“transformations”
that
are
compatible
with
the
ring
structures
on
the
domain
and
codomain,
i.e.,
morphisms
of
rings/schemes,
which
tautologically
give
rise
to
functorial
morphisms
between
the
respective
étale
fundamental
groups.
Put
another
way,
in
the
notation
of
[IUTchI],
Definition
3.1,
(e),
(f)
[which
will
be
applied
throughout
the
remainder
of
the
present
Remark
3.6.3],
for,
say,
v
∈
V
non
,
the
only
natural
correspondence
that
may
be
described
by
means
of
set-
theoretic
formulas
between
the
isomorphs
of
the
local
base
field
Ga-
lois
groups
“G
v
”
on
either
side
of
a
vertical
or
horizontal
arrow
of
the
log-theta-lattice
is
the
correspondence
constituted
by
an
indeterminate
isomorphism
of
topological
groups.
A
similar
statement
may
be
made
concerning
the
isomorphs
of
the
geometric
funda-
def
mental
group
Δ
v
=
Ker(Π
v
G
v
)
on
either
side
of
a
vertical
[but
not
horizontal!
—
cf.
the
discussion
of
(ii)
below]
arrow
of
the
log-theta-lattice
—
that
is
to
say,
the
only
natural
correspondence
that
may
described
by
means
of
set-
theoretic
formulas
between
these
isomorphs
is
the
correspondence
con-
stituted
by
an
indeterminate
isomorphism
of
topological
groups
equipped
with
some
outer
action
by
the
respective
isomorph
of
“G
v
”
—
cf.
the
discussion
of
[IUTchIII],
Remark
1.2.4.
Here,
again
we
recall
from
the
discussion
of
Remark
3.6.2,
(i),
(ii),
that
it
is
only
by
working
with
such
corre-
spondences
that
may
be
described
by
means
of
set-theoretic
formulas
that
one
may
obtain
descriptions
that
allow
one
to
calculate
the
operations
performed
in
one
universe
from
the
point
of
view
of
an
alien
universe.
(ii)
One
fundamental
difference
between
the
vertical
and
horizontal
arrows
of
the
log-theta-lattice
is
that
whereas,
for,
say,
v
∈
V
bad
,
(V1)
one
identifies,
up
to
isomorphism,
the
isomorphs
of
the
full
arithmetic
fundamental
group
“Π
v
”
on
either
side
of
a
vertical
arrow,
(H1)
one
distinguishes
the
“Δ
v
’s”
on
either
side
of
a
horizontal
arrow,
i.e.,
one
only
identifies,
up
to
isomorphism,
the
local
base
field
Galois
groups
“G
v
”
on
either
side
of
a
horizontal
arrow.
—
cf.
the
discussion
of
[IUTchIII],
Remark
1.4.2.
One
way
to
understand
the
fundamental
reason
for
this
difference
is
as
follows.
(V2)
In
order
to
construct
the
log-link
—
i.e.,
at
a
more
concrete
level,
the
power
series
that
defines
the
p
v
-adic
logarithm
at
v
—
it
is
necessary
to
avail
oneself
of
the
local
ring
structures
at
v
[cf.
the
discussion
of
[IUTchIII],
Definition
1.1,
(i),
(ii)],
which
may
only
be
reconstructed
from
84
SHINICHI
MOCHIZUKI
the
full
“Π
v
”
[i.e.,
not
from
“G
v
stripped
of
its
structure
as
a
quotient
of
Π
v
”
—
cf.
the
discussion
of
[IUTchIII],
Remark
1.4.1,
(i);
[IUTchIII],
Remark
2.1.1,
(ii);
[AbsTopIII],
§I3].
×μ
×μ
(H2)
In
order
to
construct
the
Θ
×μ
gau
-/Θ
LGP
-/Θ
lgp
-links
—
i.e.,
at
a
more
concrete
level,
the
correspondence
q
→
q
j
2
j=1,...,l
[cf.
[IUTchII],
Remark
4.11.1]
—
it
is
necessary,
in
effect,
to
construct
an
“isomorphism”
between
a
mathematical
object
[i.e.,
the
theta
values
2
“q
j
”]
that
depends,
in
an
essential
way,
on
regarding
the
various
“j”
as
distinct
labels
[which
are
constructed
from
“Δ
v
”!]
and
a
mathematical
object
[i.e.,
“q”]
that
is
independent
of
these
labels;
it
is
then
a
tautol-
ogy
that
such
an
“isomorphism”
may
only
be
achieved
if
the
labels
—
i.e.,
in
essence,
“Δ
v
”
—
on
either
side
of
the
“isomorphism”
are
kept
distinct
from
one
another.
Here,
we
observe
in
passing
that
the
“apparently
horizontal
arrow-related”
issue
dis-
cussed
in
(H2)
of
simultaneous
realization
of
“label-dependent”
and
“label-
free”
mathematical
objects
is
reminiscent
of
the
vertical
arrow
portion
of
the
bi-
coricity
theory
of
[IUTchIII],
Theorem
1.5
—
cf.
the
discussion
of
[IUTchIII],
Remark
1.5.1,
(i),
(ii);
Step
(vii)
of
the
proof
of
[IUTchIII],
Corollary
3.12.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IV
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